A057077 -id:A057077 - cp's OEIS frontend

A000009 Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts.

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296, 340, 390, 448, 512, 585, 668, 760, 864, 982, 1113, 1260, 1426, 1610, 1816, 2048, 2304, 2590, 2910, 3264, 3658, 4097, 4582, 5120, 5718, 6378

Offset: 0

N. J. A. Sloane

Partitions into distinct parts are sometimes called "strict partitions".
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The result that number of partitions of n into distinct parts = number of partitions of n into odd parts is due to Euler.
Bijection: given n = L1* 1 + L2*3 + L3*5 + L7*7 + ..., a partition into odd parts, write each Li in binary, Li = 2^a1 + 2^a2 + 2^a3 + ... where the aj's are all different, then expand n = (2^a1 * 1 + ...)*1 + ... by removing the brackets and we get a partition into distinct parts. For the reverse operation, just keep splitting any even number into halves until no evens remain.
Euler transform of period 2 sequence [1,0,1,0,...]. - Michael Somos, Dec 16 2002
Number of different partial sums 1+[1,2]+[1,3]+[1,4]+..., where [1,x] indicates a choice. E.g., a(6)=4, as we can write 1+1+1+1+1+1, 1+2+3, 1+2+1+1+1, 1+1+3+1. - Jon Perry, Dec 31 2003
a(n) is the sum of the number of partitions of x_j into at most j parts, where j is the index for the j-th triangular number and n-T(j)=x_j. For example; a(12)=partitions into <= 4 parts of 12-T(4)=2 + partitions into <= 3 parts of 12-T(3)=6 + partitions into <= 2 parts of 12-T(2)=9 + partitions into 1 part of 12-T(1)=11 = (2)(11) + (6)(51)(42)(411)(33)(321)(222) + (9)(81)(72)(63)(54)+(11) = 2+7+5+1 = 15. - Jon Perry, Jan 13 2004
Number of partitions of n where if k is the largest part, all parts 1..k are present. - Jon Perry, Sep 21 2005
Jack Grahl and Franklin T. Adams-Watters prove this claim of Jon Perry's by observing that the Ferrers dual of a "gapless" partition is guaranteed to have distinct parts; since the Ferrers dual is an involution, this establishes a bijection between the two sets of partitions. - Allan C. Wechsler, Sep 28 2021
The number of connected threshold graphs having n edges. - Michael D. Barrus (mbarrus2(AT)uiuc.edu), Jul 12 2007
Starting with offset 1 = row sums of triangle A146061 and the INVERT transform of A000700 starting: (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 4, -5, ...). - Gary W. Adamson, Oct 26 2008
Number of partitions of n in which the largest part occurs an odd number of times and all other parts occur an even number of times. (Such partitions are the duals of the partitions with odd parts.) - David Wasserman, Mar 04 2009
Equals A035363 convolved with A010054. The convolution square of A000009 = A022567 = A000041 convolved with A010054. A000041 = A000009 convolved with A035363. - Gary W. Adamson, Jun 11 2009
Considering all partitions of n into distinct parts: there are A140207(n) partitions of maximal size which is A003056(n), and A051162(n) is the greatest number occurring in these partitions. - Reinhard Zumkeller, Jun 13 2009
Equals left border of triangle A091602 starting with offset 1. - Gary W. Adamson, Mar 13 2010
Number of symmetric unimodal compositions of n+1 where the maximal part appears once. Also number of symmetric unimodal compositions of n where the maximal part appears an odd number of times. - Joerg Arndt, Jun 11 2013
Because for these partitions the exponents of the parts 1, 2, ... are either 0 or 1 (j^0 meaning that part j is absent) one could call these partitions also 'fermionic partitions'. The parts are the levels, that is the positive integers, and the occupation number is either 0 or 1 (like Pauli's exclusion principle). The 'fermionic states' are denoted by these partitions of n. - Wolfdieter Lang, May 14 2014
The set of partitions containing only odd parts forms a monoid under the product described in comments to A047993. - Richard Locke Peterson, Aug 16 2018
Ewell (1973) gives a number of recurrences. - N. J. A. Sloane, Jan 14 2020
a(n) equals the number of permutations p of the set {1,2,...,n+1}, written in one line notation as p = p_1p_2...p_(n+1), satisfying p_(i+1) - p_i <= 1 for 1 <= i <= n, (i.e., those permutations that, when read from left to right, never increase by more than 1) whose major index maj(p) := Sum_{p_i > p_(i+1)} i  equals n. For example, of the 16 permutations on 5 letters satisfying p_(i+1) - p_i <= 1, 1 <= i <= 4, there are exactly two permutations whose major index is 4, namely, 5 3 4 1 2 and 2 3 4 5 1. Hence a(4) = 2. See the Bala link in A007318 for a proof. - Peter Bala, Mar 30 2022
Conjecture: Each positive integer n can be written as a_1 + ... + a_k, where a_1,...,a_k are strict partition numbers (i.e., terms of the current sequence) with no one dividing another. This has been verified for n = 1..1350. - Zhi-Wei Sun, Apr 14 2023
Conjecture: For each integer n > 7, a(n) divides none of p(n), p(n) - 1 and p(n) + 1, where p(n) is the number of partitions of n given by A000041. This has been verified for n up to 10^5. - Zhi-Wei Sun, May 20 2023 [Verified for n <= 2*10^6. - Vaclav Kotesovec, May 23 2023]
The g.f. Product_{k >= 0} 1 + x^k = Product_{k >= 0} 1 - x^k + 2*x^k == Product_{k >= 0} 1 - x^k  == Sum_{k in Z} (-1)^k*x^(k*(3*k-1)/2) (mod 2) by Euler's pentagonal number theorem. It follows that a(n) is odd iff n = k*(3*k - 1)/2 for some integer k, i.e., iff n is a generalized pentagonal number A001318. - Peter Bala, Jan 07 2025

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 8*x^9 + ...
G.f. = q + q^25 + q^49 + 2*q^73 + 2*q^97 + 3*q^121 + 4*q^145 + 5*q^169 + ...
The partitions of n into distinct parts (see A118457) for small n are:
  1: 1
  2: 2
  3: 3, 21
  4: 4, 31
  5: 5, 41, 32
  6: 6, 51, 42, 321
  7: 7, 61, 52, 43, 421
  8: 8, 71, 62, 53, 521, 431
  ...
From _Reinhard Zumkeller_, Jun 13 2009: (Start)
a(8)=6, A140207(8)=#{5+2+1,4+3+1}=2, A003056(8)=3, A051162(8)=5;
a(9)=8, A140207(9)=#{6+2+1,5+3+1,4+3+2}=3, A003056(9)=3, A051162(9)=6;
a(10)=10, A140207(10)=#{4+3+2+1}=1, A003056(10)=4, A051162(10)=4. (End)

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Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 196.
T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, Problem 18.
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 277, Theorems 344, 346.
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Srinivasa Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. See Table V on page 309.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 288-290.

Felix Huber, Table of n, a(n) for n = 0..10000 (terms up to n=2000 from N. J. A. Sloane, up to n=5000 from Reinhard Zumkeller)
Joerg Arndt, Matters Computational (The Fxtbook), pp. 348-350.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 836.
Francesca Aicardi, Matricial formulae for partitions, Functional Analysis and Other Mathematics, Vol. 3, No. 2 (2011), pp. 127-133; arXiv preprint, arXiv:0806.1273 [math.NT], 2008.
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George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See page 4 equation (2.2).
Brennan Benfield and Arindam Roy, Log-concavity And The Multiplicative Properties of Restricted Partition Functions, arXiv:2404.03153 [math.NT], 2024.
Helena Bergold, Lukas Egeling, and Hung. P. Hoang, Signotopes with few plus signs, arXiv:2411.19208 [math.CO], 2024. See p. 14.
Andreas B. G. Blobel, An Asymptotic Form of the Generating Function Prod_{k=1,oo} (1+x^k/k), arXiv:1904.07808 [math.CO], 2019.
Alexander Bors, Michael Giudici, and Cheryl E. Praeger, Documentation for the GAP code file OrbOrd.txt, arXiv:1910.12570 [math.GR], 2019.
Henry Bottomley, Illustration for A000009, A000041, A047967.
Andrés Eduardo Caicedo and Brittany Shelton, Of puzzles and partitions: Introducing Partiti, Mathematics Magazine, Vol. 91, No. 1 (2018), pp. 20-23; arXiv preprint, arXiv:1710.04495 [math.HO], 2017.
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Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002. [Local copy, with permission]
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Evangelos Georgiadis, Computing Partition Numbers q(n), Technical Report, February (2009).
Benjamin Hackl, 5 + 5 + 1 + 1 + 1 = 10 + 2 + 1, and why there is more to it than you think., YouTube video, 2022.
Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, Journal of Number Theory, Vol. 193 (2018), pp. 171-188; arXiv preprint, arXiv:1804.09883 [math.NT], 2018.
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Donald J. Newman, A Problem Seminar, pp. 18;93;102-3 Prob. 93 Springer-Verlag NY 1982.
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Index entries for "core" sequences

Apart from the first term, equals A052839-1. The rows of A053632 converge to this sequence. When reduced modulo 2 equals the absolute values of A010815. The positions of odd terms given by A001318.
a(n) = Sum_{n=1..m} A097306(n, m), row sums of triangle of number of partitions of n into m odd parts.
Cf. A001318, A000041, A000700, A003724, A004111, A007837, A010815, A035294, A068049, A078408, A081360, A088670, A109950, A109968, A132312, A146061, A035363, A010054, A057077, A089806, A091602, A237515, A118457 (the partitions), A118459 (partition lengths), A015723 (total number of parts), A230957 (boustrophedon transform).
Cf. A167377 (complement).
Cf. A067659 (odd number of parts), A067661 (even number of parts).
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

import Data.MemoCombinators (memo2, integral)
a000009 n = a000009_list !! n
a000009_list = map (pM 1) [0..] where
   pM = memo2 integral integral p
   p _ 0 = 1
   p k m | m < k     = 0
         | otherwise = pM (k + 1) (m - k) + pM (k + 1) m
-- Reinhard Zumkeller, Sep 09 2015, Nov 05 2013

# uses A010815
using Memoize
@memoize function A000009(n)
    n == 0 && return 1
    s = sum((-1)^k*A000009(n - k^2) for k in 1:isqrt(n))
    A010815(n) - 2*s
end # Peter Luschny, Sep 09 2021

Coefficients(&*[1+x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

N := 100; t1 := series(mul(1+x^k,k=1..N),x,N); A000009 := proc(n) coeff(t1,x,n); end;
spec := [ P, {P=PowerSet(N), N=Sequence(Z,card>=1)} ]: [ seq(combstruct[count](spec, size=n), n=0..58) ];
spec := [ P, {P=PowerSet(N), N=Sequence(Z,card>=1)} ]: combstruct[allstructs](spec, size=10); # to get the actual partitions for n=10
A000009 := proc(n)
    local x,m;
    product(1+x^m,m=1..n+1) ;
    expand(%) ;
    coeff(%,x,n) ;
end proc: # R. J. Mathar, Jun 18 2016
lim := 99; # Enlarge if more terms are needed.
simplify(expand(QDifferenceEquations:-QPochhammer(-1, x, lim)/2, x)):
seq(coeff(%, x, n), n=0..55); # Peter Luschny, Nov 17 2016
# Alternative:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..55);  # Alois P. Heinz, Jun 24 2025

PartitionsQ[Range[0, 60]] (* Harvey Dale, Jul 27 2009 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 06 2011 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k, {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Jul 06 2011 *)
a[ n_] := With[ {t = Log[q] / (2 Pi I)}, SeriesCoefficient[ q^(-1/24) DedekindEta[2 t] / DedekindEta[ t], {q, 0, n}]]; (* Michael Somos, Jul 06 2011 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, May 24 2013 *)
a[ n_] := SeriesCoefficient[ Series[ QHypergeometricPFQ[ {q}, {q x}, q, - q x], {q, 0, n}] /. x -> 1, {q, 0, n}]; (* Michael Somos, Mar 04 2014 *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[{}, {}, q, -1] / 2, {q, 0, n}]; (* Michael Somos, Mar 04 2014 *)
nmax = 60; CoefficientList[Series[Exp[Sum[(-1)^(k+1)/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 14 2017 *)

num_distinct_partitions(60,list); /* Emanuele Munarini, Feb 24 2014 */

h(n):=if oddp(n)=true then 1 else 0;
S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Nov 17 1999 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))};

{a(n) = my(c); forpart(p=n, if( n<1 || p[1]<2, c++; for(i=1, #p-1, if( p[i+1] > p[i]+1, c--; break)))); c}; /* Michael Somos, Aug 13 2017 */

lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)/eta(q))} \\ Altug Alkan, Mar 20 2018

# uses A010815
from functools import lru_cache
from math import isqrt
@lru_cache(maxsize=None)
def A000009(n): return 1 if n == 0 else A010815(n)+2*sum((-1)**(k+1)*A000009(n-k**2) for k in range(1,isqrt(n)+1)) # Chai Wah Wu, Sep 08 2021

import numpy as np
n = 1000
arr = np.zeros(n,dtype=object)
arr[0] = 1
for i in range(1,n):
    arr[i:] += arr[:n-i]
print(arr) # Yigit Oktar, Jul 12 2025

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1)
b = EulerTransform(a)
print([b(n) for n in range(56)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{m>=1} (1 + x^m) = 1/Product_{m>=0} (1-x^(2m+1)) = Sum_{k>=0} Product_{i=1..k} x^i/(1-x^i) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1-x^k).
G.f.: Sum_{n>=0} x^n*Product_{k=1..n-1} (1+x^k) = 1 + Sum_{n>=1} x^n*Product_{k>=n+1} (1+x^k). - Joerg Arndt, Jan 29 2011
Product_{k>=1} (1+x^(2k)) = Sum_{k>=0} x^(k*(k+1))/Product_{i=1..k} (1-x^(2i)) - Euler (Hardy and Wright, Theorem 346).
Asymptotics: a(n) ~ exp(Pi l_n / sqrt(3)) / ( 4 3^(1/4) l_n^(3/2) ) where l_n = (n-1/24)^(1/2) (Ayoub).
For n > 1, a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), with a(0)=1, b(n) = A000593(n) = sum of odd divisors of n; cf. A000700. - Vladeta Jovovic, Jan 21 2002
a(n) = t(n, 0), t as defined in A079211.
a(n) = Sum_{k=0..n-1} A117195(n,k) = A117192(n) + A117193(n) for n>0. - Reinhard Zumkeller, Mar 03 2006
a(n) = A026837(n) + A026838(n) = A118301(n) + A118302(n); a(A001318(n)) = A051044(n); a(A090864(n)) = A118303(n). - Reinhard Zumkeller, Apr 22 2006
Expansion of 1 / chi(-x) = chi(x) / chi(-x^2) = f(-x) / phi(x) = f(x) / phi(-x^2) = psi(x) / f(-x^2) = f(-x^2) / f(-x) = f(-x^4) / psi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Mar 12 2011
G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = 2^(-1/2) / f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 16 2007
Expansion of q^(-1/24) * eta(q^2) / eta(q) in powers of q.
Expansion of q^(-1/24) 2^(-1/2) f2(t) in powers of q = exp(2 Pi i t) where f2() is a Weber function. - Michael Somos, Oct 18 2007
Given g.f. A(x), then B(x) = x * A(x^3)^8 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = v - u^2 + 16*u*v^2 . - Michael Somos, May 31 2005
Given g.f. A(x), then B(x) = x * A(x^8)^3 satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (u^3 - v) * (u + v^3) - 9 * u^3 * v^3. - Michael Somos, Mar 25 2008
From Evangelos Georgiadis, Andrew V. Sutherland, Kiran S. Kedlaya (egeorg(AT)mit.edu), Mar 03 2009: (Start)
a(0)=1; a(n) = 2*(Sum_{k=1..floor(sqrt(n))} (-1)^(k+1) a(n-k^2)) + sigma(n) where sigma(n) = (-1)^j if (n=(j*(3*j+1))/2 OR n=(j*(3*j-1))/2) otherwise sigma(n)=0 (simpler: sigma = A010815). (End)
From Gary W. Adamson, Jun 13 2009: (Start)
The product g.f. = (1/(1-x))*(1/(1-x^3))*(1/(1-x^5))*...; = (1,1,1,...)*
(1,0,0,1,0,0,1,0,0,1,...)*(1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,...) * ...; =
a*b*c*... where a, a*b, a*b*c, ... converge to A000009:
1, 1, 1, 2, 2, 2, 3, 3, 3, 4, ... = a*b
1, 1, 1, 2, 2, 3, 4, 4, 5, 6, ... = a*b*c
1, 1, 1, 2, 2, 3, 4, 5, 6, 7, ... = a*b*c*d
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, ... = a*b*c*d*e
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, ... = a*b*c*d*e*f
... (cf. analogous example in A000041). (End)
a(A004526(n)) = A172033(n). - Reinhard Zumkeller, Jan 23 2010
a(n) = P(n) - P(n-2) - P(n-4) + P(n-10) + P(n-14) + ... + (-1)^m P(n-2p_m) + ..., where P(n) is the partition function (A000041) and p_m = m(3m-1)/2 is the m-th generalized pentagonal number (A001318). - Jerome Malenfant, Feb 16 2011
a(n) = A054242(n,0) = A201377(n,0). - Reinhard Zumkeller, Dec 02 2011
G.f.: 1/2 (-1; x){inf} where (a; q)_k is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Apr 24 2013
More precise asymptotics: a(n) ~ exp(Pi*sqrt((n-1/24)/3)) / (4*3^(1/4)*(n-1/24)^(3/4)) * (1 + (Pi^2-27)/(24*Pi*sqrt(3*(n-1/24))) + (Pi^4-270*Pi^2-1215)/(3456*Pi^2*(n-1/24))). - Vaclav Kotesovec, Nov 30 2015
a(n) = A067661(n) + A067659(n). Wolfdieter Lang, Jan 18 2016
From Vaclav Kotesovec, May 29 2016: (Start)
a(n) ~ exp(Pi*sqrt(n/3))/(4*3^(1/4)*n^(3/4)) * (1 + (Pi/(48*sqrt(3)) - (3*sqrt(3))/(8*Pi))/sqrt(n) + (Pi^2/13824 - 5/128 - 45/(128*Pi^2))/n).
a(n) ~ exp(Pi*sqrt(n/3) + (Pi/(48*sqrt(3)) - 3*sqrt(3)/(8*Pi))/sqrt(n) - (1/32 + 9/(16*Pi^2))/n) / (4*3^(1/4)*n^(3/4)).
(End)
a(n) = A089806(n)*A010815(floor(n/2)) + a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) + ... + A057077(m-1)*a(n-A001318(m)) + ..., where n > A001318(m). - Gevorg Hmayakyan, Jul 07 2016
a(n) ~ Pi*BesselI(1, Pi*sqrt((n+1/24)/3)) / sqrt(24*n+1). - Vaclav Kotesovec, Nov 08 2016
a(n) = A000041(n) - A047967(n). - R. J. Mathar, Nov 20 2017
Sum_{n>=1} 1/a(n) = A237515. - Amiram Eldar, Nov 15 2020
From Peter Bala, Jan 15 2021: (Start)
G.f.: (1 + x)*Sum_{n >= 0} x^(n*(n+3)/2)/Product_{k = 1..n} (1 - x^k) =
(1 + x)*(1 + x^2)*Sum_{n >= 0} x^(n*(n+5)/2)/Product_{k = 1..n} (1 - x^k) = (1 + x)*(1 + x^2)*(1 + x^3)*Sum_{n >= 0} x^(n*(n+7)/2)/Product_{k = 1..n} (1 - x^k) = ....
G.f.: (1/2)*Sum_{n >= 0} x^(n*(n-1)/2)/Product_{k = 1..n} (1 - x^k) =
(1/2)*(1/(1 + x))*Sum_{n >= 0} x^((n-1)*(n-2)/2)/Product_{k = 1..n} (1 - x^k) = (1/2)*(1/((1 + x)*(1 + x^2)))*Sum_{n >= 0} x^((n-2)*(n-3)/2)/Product_{k = 1..n} (1 - x^k) = ....
G.f.: Sum_{n >= 0} x^n/Product_{k = 1..n} (1 - x^(2*k)) = (1/(1 - x)) * Sum_{n >= 0} x^(3*n)/Product_{k = 1..n} (1 - x^(2*k)) =  (1/((1 - x)*(1 - x^3))) * Sum_{n >= 0} x^(5*n)/Product_{k = 1..n} (1 - x^(2*k)) = (1/((1 - x)*(1 - x^3)*(1 - x^5))) * Sum_{n >= 0} x^(7*n)/Product_{k = 1..n} (1 - x^(2*k)) = .... (End)
From Peter Bala, Feb 02 2021: (Start)
G.f.: A(x) = Sum_{n >= 0} x^(n*(2*n-1))/Product_{k = 1..2*n} (1 - x^k). (Set z = x and q = x^2 in Mc Laughlin et al. (2019 ArXiv version), Section 1.3, Identity 7.)
Similarly, A(x) = Sum_{n >= 0} x^(n*(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k). (End)
a(n) = A001227(n) + A238005(n) + A238006(n). - R. J. Mathar, Sep 08 2021
G.f.: A(x) = exp ( Sum_{n >= 1} x^n/(n*(1 - x^(2*n))) ) = exp ( Sum_{n >= 1} (-1)^(n+1)*x^n/(n*(1 - x^n)) ). - Peter Bala, Dec 23 2021
Sum_{n>=0} a(n)/exp(Pi*n) = exp(Pi/24)/2^(1/8) = A292820. - Simon Plouffe, May 12 2023 [Proof: Sum_{n>=0} a(n)/exp(Pi*n) = phi(exp(-2*Pi)) / phi(exp(-Pi)), where phi(q) is the Euler modular function. We have phi(exp(-2*Pi)) = exp(Pi/12) * Gamma(1/4) / (2 * Pi^(3/4)) and phi(exp(-Pi)) = exp(Pi/24) * Gamma(1/4) / (2^(7/8) * Pi^(3/4)), see formulas (14) and (13) in I. Mező, 2013. - Vaclav Kotesovec, May 12 2023]
a(2*n) = Sum_{j=1..n} p(n+j, 2*j) and a(2*n+1) = Sum_{j=1..n+1} p(n+j,2*j-1), where p(n, s) is the number of partitions of n having exactly s parts. - Gregory L. Simay, Aug 30 2023

A007814 Exponent of highest power of 2 dividing n, a.k.a. the binary carry sequence, the ruler sequence, or the 2-adic valuation of n.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0

Offset: 1

John Tromp, Dec 11 1996

This sequence is an exception to my usual rule that when every other term of a sequence is 0 then those 0's should be omitted. In this case we would get A001511. - N. J. A. Sloane
To construct the sequence: start with 0,1, concatenate to get 0,1,0,1. Add + 1 to last term gives 0,1,0,2. Concatenate those 4 terms to get 0,1,0,2,0,1,0,2. Add + 1 to last term etc. - Benoit Cloitre, Mar 06 2003
The sequence is invariant under the following two transformations: increment every element by one (1, 2, 1, 3, 1, 2, 1, 4, ...), put a zero in front and between adjacent elements (0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ...). The intermediate result is A001511. - Ralf Hinze (ralf(AT)informatik.uni-bonn.de), Aug 26 2003
Fixed point of the morphism 0->01, 1->02, 2->03, 3->04, ..., n->0(n+1), ..., starting from a(1) = 0. - Philippe Deléham, Mar 15 2004
Fixed point of the morphism 0->010, 1->2, 2->3, ..., n->(n+1), .... - Joerg Arndt, Apr 29 2014
a(n) is also the number of times to repeat a step on an even number in the hailstone sequence referenced in the Collatz conjecture. - Alex T. Flood (whiteangelsgrace(AT)gmail.com), Sep 22 2006
Let F(n) be the n-th Fermat number (A000215). Then F(a(r-1)) divides F(n)+2^k for r = k mod 2^n and r != 1. - T. D. Noe, Jul 12 2007
The following relation holds: 2^A007814(n)*(2*A025480(n-1)+1) = A001477(n) = n. (See functions hd, tl and cons in [Paul Tarau 2009].)
a(n) is the number of 0's at the end of n when n is written in base 2.
a(n+1) is the number of 1's at the end of n when n is written in base 2. - M. F. Hasler, Aug 25 2012
Shows which bit to flip when creating the binary reflected Gray code (bits are numbered from the right, offset is 0). That is, A003188(n) XOR A003188(n+1) == 2^A007814(n). - Russ Cox, Dec 04 2010
The sequence is squarefree (in the sense of not containing any subsequence of the form XX) [Allouche and Shallit]. Of course it contains individual terms that are squares (such as 4). - Comment expanded by N. J. A. Sloane, Jan 28 2019
a(n) is the number of zero coefficients in the n-th Stern polynomial, A125184. - T. D. Noe, Mar 01 2011
Lemma: For n < m with r = a(n) = a(m) there exists n < k < m with a(k) > r. Proof: We have n=b2^r and m=c2^r with b < c both odd; choose an even i between them; now a(i2^r) > r and n < i2^r < m. QED. Corollary: Every finite run of consecutive integers has a unique maximum 2-adic valuation. - Jason Kimberley, Sep 09 2011
a(n-2) is the 2-adic valuation of A000166(n) for n >= 2. - Joerg Arndt, Sep 06 2014
a(n) = number of 1's in the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} p_j-th prime (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(24)=3; indeed, the partition having Heinz number 24 = 2*2*2*3  is [1,1,1,2]. - Emeric Deutsch, Jun 04 2015
a(n+1) is the difference between the two largest parts in the integer partition having viabin number n (0 is assumed to be a part). Example: a(20) = 2. Indeed, we have 19 = 10011_2, leading to the Ferrers board of the partition [3,1,1]. For the definition of viabin number see the comment in A290253. - Emeric Deutsch, Aug 24 2017
Apart from being squarefree, as noted above, the sequence has the property that every consecutive subsequence contains at least one number an odd number of times. - Jon Richfield, Dec 20 2018
a(n+1) is the 2-adic valuation of Sum_{e=0..n} u^e = (1 + u + u^2 + ... + u^n), for any u of the form 4k+1 (A016813). - Antti Karttunen, Aug 15 2020
{a(n)} represents the "first black hat" strategy for the game of countably infinitely many hats, with a probability of success of 1/3; cf. the Numberphile link below. - Frederic Ruget, Jun 14 2021
a(n) is the least nonnegative integer k for which there does not exist i+j=n and a(i)=a(j)=k (cf. A322523). - Rémy Sigrist and Jianing Song, Aug 23 2022

			2^3 divides 24, so a(24)=3.
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
  0;
  1,0;
  2,0,1,0;
  3,0,1,0,2,0,1,0;
  4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
  5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
  6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,...
(End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 27.
K. Atanassov, On the 37th and the 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

T. D. Noe, Table of n, a(n) for n = 1..10000
Joerg Arndt, Subset-lex: did we miss an order?, arXiv:1405.6503 [math.CO], 2014.
K. Atanassov, On Some of Smarandache's Problems
Alain Connes, Caterina Consani, and Henri Moscovici, Zeta zeros and prolate wave operators, arXiv:2310.18423 [math.NT], 2023.
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, (2009) Discrete Math., 309 (2009), 6245-6254.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, arXiv:0901.1397 [math.CO], 2009.
M. Hassani, Equations and inequalities involving v_p(n!), J. Inequ. Pure Appl. Math. 6 (2005) vol. 2, #29.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 61. Book's website
R. Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 3.2.3.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160.
Francis Laclé, 2-adic parity explorations of the 3n+ 1 problem, hal-03201180v2 [cs.DM], 2021.
Shuo Li, Palindromic length sequence of the ruler sequence and of the period-doubling sequence, arXiv:2007.08317 [math.CO], 2020.
Nicolas Mallet, Trial for a proof of the Syracuse conjecture, arXiv preprint arXiv:1507.05039 [math.GM], 2015.
S. Mazzanti, Plain Bases for Classes of Primitive Recursive Functions, Mathematical Logic Quarterly, 48 (2002).
Matthew Andres Moreno, Luis Zaman, and Emily Dolson, Structured Downsampling for Fast, Memory-efficient Curation of Online Data Streams, arXiv:2409.06199 [cs.DS], 2024. See p. 5.
Sascha Mücke, Coding Nuggets Faster QUBO Brute-Force Solving, TU Dortmund Univ. (Germany 2023).
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
Numberphile, Hat Problems - Numberphile
Giovanni Pighizzini, Limited Automata: Properties, Complexity and Variants, International Conference on Descriptional Complexity of Formal Systems (DCFS 2019) Descriptional Complexity of Formal Systems, Lecture Notes in Computer Science (LNCS, Vol. 11612) Springer, Cham, 57-73.
Simon Plouffe, On the values of the functions ... [zeta and Gamma] ..., arXiv preprint arXiv:1310.7195 [math.NT], 2013.
A. Postnikov (MIT) and B. Sagan, What power of two divides a weighted Catalan number?, arXiv:math/0601339 [math.CO], 2006.
Lara Pudwell and Eric Rowland, Avoiding fractional powers over the natural numbers, arXiv:1510.02807 [math.CO] (2015). Also Electronic Journal of Combinatorics, Volume 25(2) (2018), #P2.27. See Section 2.
Ville Salo, Decidability and Universality of Quasiminimal Subshifts, arXiv preprint arXiv:1411.6644 [math.DS], 2014.
Vladimir Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2014.
F. Smarandache, Only Problems, Not Solutions!.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Paul Tarau, A Groupoid of Isomorphic Data Transformations. Calculemus 2009, 8th International Conference, MKM 2009, pp. 170-185, Springer, LNAI 5625.
P. M. B. Vitanyi, An optimal simulation of counter machines, SIAM J. Comput, 14:1(1985), 1-33.
Eric Weisstein's World of Mathematics, Binary, Binary Carry Sequence, and Double-Free Set.
Wikipedia, P-adic order.
Index entries for sequences that are fixed points of mappings
Index entries for sequences related to binary expansion of n

Cf. A011371 (partial sums), A094267 (first differences), A001511 (bisection), A346070 (mod 4).
Bisection of A050605 and |A088705|. Pairwise sums are A050603 and A136480. Difference of A285406 and A281264.
This is Guy Steele's sequence GS(1, 4) (see A135416). Cf. A053398(1,n). Column/row 1 of table A050602.
Cf. A007949 (3-adic), A235127 (4-adic), A112765 (5-adic), A122841 (6-adic), A214411 (7-adic), A244413 (8-adic), A122840 (10-adic).
Cf. A000079, A002487, A006519, A051064, A055457, A063787, A215366, A220466.
Cf. A086463 (Dgf at s=2).

a007814 n = if m == 0 then 1 + a007814 n' else 0
            where (n', m) = divMod n 2
-- Reinhard Zumkeller, Jul 05 2013, May 14 2011, Apr 08 2011

a007814 n | odd n = 0 | otherwise = 1 + a007814 (n `div` 2)
--  Walt Rorie-Baety, Mar 22 2013

[Valuation(n, 2): n in [1..120]]; // Bruno Berselli, Aug 05 2013

ord := proc(n) local i,j; if n=0 then return 0; fi; i:=0; j:=n; while j mod 2 <> 1 do i:=i+1; j:=j/2; od: i; end proc: seq(ord(n), n=1..111);
A007814 := n -> padic[ordp](n,2): seq(A007814(n), n=1..111); # Peter Luschny, Nov 26 2010

Table[IntegerExponent[n, 2], {n, 64}] (* Eric W. Weisstein *)
IntegerExponent[Range[64], 2] (* Eric W. Weisstein, Feb 01 2024 *)
p=2; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 96 ]
DigitCount[BitXor[x, x - 1], 2, 1] - 1; a different version based on the same concept: Floor[Log[2, BitXor[x, x - 1]]] (* Jaume Simon Gispert (jaume(AT)nuem.com), Aug 29 2004 *)
Nest[Join[ #, ReplacePart[ #, Length[ # ] -> Last[ # ] + 1]] &, {0, 1}, 5] (* N. J. Gunther, May 23 2009 *)
Nest[ Flatten[# /. a_Integer -> {0, a + 1}] &, {0}, 7] (* Robert G. Wilson v, Jan 17 2011 *)

PARI
```
A007814(n)=valuation(n,2);
```

import math
def a(n): return int(math.log(n - (n & n - 1), 2)) # Indranil Ghosh, Apr 18 2017

def A007814(n): return (~n & n-1).bit_length() # Chai Wah Wu, Jul 01 2022

sapply(1:100,function(x) sum(gmp::factorize(x)==2)) # Christian N. K. Anderson, Jun 20 2013

(define (A007814 n) (let loop ((n n) (e 0)) (if (odd? n) e (loop (/ n 2) (+ 1 e))))) ;; Antti Karttunen, Oct 06 2017

a(n) = A001511(n) - 1.
a(2*n) = A050603(2*n) = A001511(n).
a(n) = A091090(n-1) + A036987(n-1) - 1.
a(n) = 0 if n is odd, otherwise 1 + a(n/2). - Reinhard Zumkeller, Aug 11 2001
Sum_{k=1..n} a(k) = n - A000120(n). - Benoit Cloitre, Oct 19 2002
G.f.: A(x) = Sum_{k>=1} x^(2^k)/(1-x^(2^k)). - Ralf Stephan, Apr 10 2002
G.f. A(x) satisfies A(x) = A(x^2) + x^2/(1-x^2). A(x) = B(x^2) = B(x) - x/(1-x), where B(x) is the g.f. for A001151. - Franklin T. Adams-Watters, Feb 09 2006
Totally additive with a(p) = 1 if p = 2, 0 otherwise.
Dirichlet g.f.: zeta(s)/(2^s-1). - Ralf Stephan, Jun 17 2007
Define 0 <= k <= 2^n - 1; binary: k = b(0) + 2*b(1) + 4*b(2) + ... + 2^(n-1)*b(n-1); where b(x) are 0 or 1 for 0 <= x <= n - 1; define c(x) = 1 - b(x) for 0 <= x <= n - 1; Then: a(k) = c(0) + c(0)*c(1) + c(0)*c(1)*c(2) + ... + c(0)*c(1)...c(n-1); a(k+1) = b(0) + b(0)*b(1) + b(0)*b(1)*b(2) + ... + b(0)*b(1)...b(n-1). - Arie Werksma (werksma(AT)tiscali.nl), May 10 2008
a(n) = floor(A002487(n - 1) / A002487(n)). - Reikku Kulon, Oct 05 2008
Sum_{k=1..n} (-1)^A000120(n-k)*a(k) = (-1)^(A000120(n)-1)*(A000120(n) - A000035(n)). - Vladimir Shevelev, Mar 17 2009
a(A001147(n) + A057077(n-1)) = a(2*n). - Vladimir Shevelev, Mar 21 2009
For n>=1, a(A004760(n+1)) = a(n). - Vladimir Shevelev, Apr 15 2009
2^(a(n)) = A006519(n). - Philippe Deléham, Apr 22 2009
a(n) = A063787(n) - A000120(n). - Gary W. Adamson, Jun 04 2009
a(C(n,k)) = A000120(k) + A000120(n-k) - A000120(n). - Vladimir Shevelev, Jul 19 2009
a(n!) = n - A000120(n). - Vladimir Shevelev, Jul 20 2009
v_{2}(n) = Sum_{r>=1} (r / 2^(r+1)) Sum_{k=0..2^(r+1)-1} e^(2(k*Pi*i(n+2^r))/(2^(r+1))). - A. Neves, Sep 28 2010, corrected Oct 04 2010
a(n) mod 2 = A096268(n-1). - Robert G. Wilson v, Jan 18 2012
a(A005408(n)) = 1; a(A016825(n)) = 3; A017113(a(n)) = 5; A051062(a(n)) = 7; a(n) = (A037227(n)-1)/2. - Reinhard Zumkeller, Jun 30 2012
a((2*n-1)*2^p) = p, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 04 2013
a(n) = A067255(n,1). - Reinhard Zumkeller, Jun 11 2013
a(n) = log_2(n - (n AND n-1)). - Gary Detlefs, Jun 13 2014
a(n) = 1 + A000120(n-1) - A000120(n), where A000120 is the Hamming weight function. - Stanislav Sykora, Jul 14 2014
A053398(n,k) = a(A003986(n-1,k-1)+1); a(n) = A053398(n,1) = A053398(n,n) = A053398(2*n-1,n) = Min_{k=1..n} A053398(n,k). - Reinhard Zumkeller, Aug 04 2014
a((2*x-1)*2^n) = a((2*y-1)*2^n) for positive n, x and y. - Juri-Stepan Gerasimov, Aug 04 2016
a(n) = A285406(n) - A281264(n). - Ralf Steiner, Apr 18 2017
a(n) = A000005(n)/(A000005(2*n) - A000005(n)) - 1. - conjectured by Velin Yanev, Jun 30 2017, proved by Nicholas Stearns, Sep 11 2017
Equivalently to above formula, a(n) = A183063(n) / A001227(n), i.e., a(n) is the number of even divisors of n divided by number of odd divisors of n. - Franklin T. Adams-Watters, Oct 31 2018
a(n)*(n mod 4) = 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Feb 16 2019
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Jul 11 2020
a(n) = 2*Sum_{j=1..floor(log_2(n))} frac(binomial(n, 2^j)*2^(j-1)/n). - Dario T. de Castro, Jul 08 2022
a(n) = A070939(n) - A070939(A030101(n)). - Andrew T. Porter, Dec 16 2022
a(n) = floor((gcd(n, 2^n)^(n+1) mod (2^(n+1)-1)^2)/(2^(n+1)-1)) (see Lemma 3.4 from Mazzanti's 2002 article). - Lorenzo Sauras Altuzarra, Mar 10 2024
a(n) = 1 - A088705(n). - Chai Wah Wu, Sep 18 2024

Formula index adapted to the offset of A025480 by R. J. Mathar, Jul 20 2010

Edited by Ralf Stephan, Feb 08 2014

A000312 a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

Original entry on oeis.org

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000, 285311670611, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, 18446744073709551616, 827240261886336764177, 39346408075296537575424, 1978419655660313589123979

Offset: 0

N. J. A. Sloane

Also number of labeled pointed rooted trees (or vertebrates) on n nodes.
For n >= 1 a(n) is also the number of n X n (0,1) matrices in which each row contains exactly one entry equal to 1. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
Also the number of labeled rooted trees on (n+1) nodes such that the root is lower than its children. Also the number of alternating labeled rooted ordered trees on (n+1) nodes such that the root is lower than its children. - Cedric Chauve (chauve(AT)lacim.uqam.ca), Mar 27 2002
With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j, i) = the j-th part of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i=1..p(n)} (n!/(Product_{j=1..p(i)} p(i, j)!)) * ((n!/(n - p(i)))!/(Product_{j=1..d(i)} m(i, j)!)). - Thomas Wieder, May 18 2005
All rational solutions to the equation x^y = y^x, with x < y, are given by x = A000169(n+1)/A000312(n), y = A000312(n+1)/A007778(n), where n = 1, 2, 3, ... . - Nick Hobson, Nov 30 2006
a(n) is the total number of leaves in all (n+1)^(n-1) trees on {0,1,2,...,n} rooted at 0. For example, with edges directed away from the root, the trees on {0,1,2} are {0->1,0->2},{0->1->2},{0->2->1} and contain a total of a(2)=4 leaves. - David Callan, Feb 01 2007
Limit_{n->infinity} A000169(n+1)/a(n) = exp(1). Convergence is slow, e.g., it takes n > 74 to get one decimal place correct and n > 163 to get two of them. - Alonso del Arte, Jun 20 2011
Also smallest k such that binomial(k, n) is divisible by n^(n-1), n > 0. - Michel Lagneau, Jul 29 2013
For n >= 2 a(n) is represented in base n as "one followed by n zeros". - R. J. Cano, Aug 22 2014
Number of length-n words over the alphabet of n letters. - Joerg Arndt, May 15 2015
Number of prime parking functions of length n+1. - Rui Duarte, Jul 27 2015
The probability density functions p(x, m=q, n=q, mu=1) = A000312(q)*E(x, q, q) and p(x, m=q, n=1, mu=q) = (A000312(q)/A000142(q-1))*x^(q-1)*E(x, q, 1), with q >= 1, lead to this sequence, see A163931, A274181 and A008276. - Johannes W. Meijer, Jun 17 2016
Satisfies Benford's law [Miller, 2015]. - N. J. A. Sloane, Feb 12 2017
A signed version of this sequence apart from the first term (1, -4, -27, 256, 3125, -46656, ...), has the following property: for every prime p == 1 (mod 2n), (-1)^(n(n-1)/2)*n^n = A057077(n)*a(n) is always a 2n-th power residue modulo p. - Jianing Song, Sep 05 2018
From Juhani Heino, May 07 2019: (Start)
n^n is both Sum_{i=0..n} binomial(n,i)*(n-1)^(n-i)
  and Sum_{i=0..n} binomial(n,i)*(n-1)^(n-i)*i.
The former is the familiar binomial distribution of a throw of n n-sided dice, according to how many times a required side appears, 0 to n. The latter is the same but each term is multiplied by its amount. This means that if the bank pays the player 1 token for each die that has the chosen side, it is always a fair game if the player pays 1 token to enter - neither bank nor player wins on average.
Examples:
2-sided dice (2 coins): 4 = 1 + 2 + 1 = 1*0 + 2*1 + 1*2 (0 omitted from now on);
3-sided dice (3 long triangular prisms): 27 = 8 + 12 + 6 + 1 = 12*1 + 6*2 + 1*3;
4-sided dice (4 long square prisms or 4 tetrahedrons): 256 = 81 + 108 + 54 + 12 + 1 = 108*1 + 54*2 + 12*3 + 1*4;
5-sided dice (5 long pentagonal prisms): 3125 = 1024 + 1280 + 640 + 160 + 20 + 1 = 1280*1 + 640*2 + 160*3 + 20*4 + 1*5;
6-sided dice (6 cubes): 46656 = 15625 + 18750 + 9375 + 2500 + 375 + 30 + 1 = 18750*1 + 9375*2 + 2500*3 + 375*4 + 30*5 + 1*6.
(End)
For each n >= 1 there is a graph on a(n) vertices whose largest independent set has size n and whose independent set sequence is constant (specifically, for each k=1,2,...,n, the graph has n^n independent sets of size k). There is no graph of smaller order with this property (Ball et al. 2019). - David Galvin, Jun 13 2019
For n >= 2 and 1 <= k <= n, a(n)*(n + 1)/4 + a(n)*(k - 1)*(n + 1 - k)/2*n is equal to the sum over all words w = w(1)...w(n) of length n over the alphabet {1, 2, ..., n} of the following quantity: Sum_{i=1..w(k)} w(i). Inspired by Problem 12432 in the AMM (see links). - Sela Fried, Dec 10 2023
Also, dimension of the unique cohomology group of the smallest interval containing the poset of partitions decorated by Perm, i.e. the poset of pointed partitions. - Bérénice Delcroix-Oger, Jun 25 2025

			G.f. = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + 3125*x^5 + 46656*x^6 + 823543*x^7 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 62, 63, 87.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 173, #39.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.37)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Kenny Lau, Table of n, a(n) for n = 0..385 [First 100 terms computed by T. D. Noe]
Taylor Ball, David Galvin, Katie Hyry, and Kyle Weingartner, Independent set and matching permutations, arXiv:1901.06579 [math.CO], 2019.
Arthur T. Benjamin and Fritz Juhnke, Another way of counting n^n, SIAM J. Discrete Math., Vol. 5, No. 3 (1992), pp. 377-379. - _N. J. A. Sloane_, Jun 09 2011
H. Bottomley, Illustration of initial terms.
H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, The Mathematical Intelligencer, Vol. 20 (4), 1998, pp. 25-29. (Sequence appears as formula in Eq. (8))
C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 34.
Frank Ellermann, Illustration of binomial transforms.
José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, Bulletin of the Korean Mathematical Society, Vol. 51, No. 5 (2014), pp. 1325-1337; arXiv preprint, arXiv:1203.4066 [math.NT], 2012.
Nick Hobson, Solution to puzzle 48: Exponential equation.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 36.
Steven J. Miller (ed.), Exercises to "The Theory and Applications of Benford's Law", Princeton University Press, 2015.
Mustafa Obaid et al., The number of complete exceptional sequences for a Dynkin algebra, arXiv preprint arXiv:1307.7573 [math.RT], 2013.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
E. Vigren (Proposer), Problem 12432, Amer. Math. Monthly 130 (2023), p. 953.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Eric Weisstein's World of Mathematics, Hadamard's Maximum Determinant Problem.
Eric Weisstein's World of Mathematics, Hankel Matrix.
Dimitri Zvonkine, An algebra of power series..., arXiv:math/0403092 [math.AG], 2004.
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to Benford's law

Cf. A000107, A000169, A000272, A001372, A007778, A007830, A008785-A008791, A019538, A048993, A008279, A085741, A062206, A212333.
First column of triangle A055858. Row sums of A066324.
Cf. A001923 (partial sums), A002109 (partial products), A007781 (first differences), A066588 (sum of digits).
Cf. A056665, A081721, A130293, A168658, A275549-A275558 (various classes of endofunctions).
Cf. A174824, A204688.
Cf. A055137, A083648.

a000312 n = n ^ n
a000312_list = zipWith (^) [0..] [0..]  -- Reinhard Zumkeller, Jul 07 2012

A000312 := n->n^n: seq(A000312(n), n=0..17);

Array[ #^# &, 16] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Table[Sum[StirlingS2[n, i] i! Binomial[n, i], {i, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, Mar 17 2009 *)
a[ n_] := If[ n < 1, Boole[n == 0], n^n]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (1 + LambertW[-x]), {x, 0, n}]]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[n < 0, 0, n! SeriesCoefficient[ Nest[ 1 / (1 - x / (1 - Integrate[#, x])) &, 1 + O[x], n], {x, 0, n}]]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[ n < 0, 0, With[{m = n + 1}, m! SeriesCoefficient[ InverseSeries[ Series[ (x - 1) Log[1 - x], {x, 0, m}]], m]]]; (* Michael Somos, May 24 2014 *)

A000312[n]:=if n=0 then 1 else n^n$
makelist(A000312[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

PARI
```
{a(n) = n^n};
```

is(n)=my(b,k=ispower(n,,&b));if(k,for(e=1,valuation(k,b), if(k/b^e == e, return(1)))); n==1 \\ Charles R Greathouse IV, Jan 14 2013

{a(n) = my(A = 1 + O(x)); if( n<0, 0, for(k=1, n, A = 1 / (1 - x / (1 - intformal( A)))); n! * polcoeff( A, n))}; /* Michael Somos, May 24 2014 */

def A000312(n): return n**n # Chai Wah Wu, Nov 07 2022

a(n-1) = -Sum_{i=1..n} (-1)^i*i*n^(n-1-i)*binomial(n, i). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
E.g.f.: 1/(1 + W(-x)), W(x) = principal branch of Lambert's function.
a(n) = Sum_{k>=0} binomial(n, k)*Stirling2(n, k)*k! = Sum_{k>=0} A008279(n,k)*A048993(n,k) = Sum_{k>=0} A019538(n,k)*A007318(n,k). - Philippe Deléham, Dec 14 2003
E.g.f.: 1/(1 - T), where T = T(x) is Euler's tree function (see A000169).
a(n) = A000169(n+1)*A128433(n+1,1)/A128434(n+1,1). - Reinhard Zumkeller, Mar 03 2007
Comment on power series with denominators a(n): Let f(x) = 1 + Sum_{n>=1} x^n/n^n. Then as x -> infinity, f(x) ~ exp(x/e)*sqrt(2*Pi*x/e). - Philippe Flajolet, Sep 11 2008
E.g.f.: 1 - exp(W(-x)) with an offset of 1 where W(x) = principal branch of Lambert's function. - Vladimir Kruchinin, Sep 15 2010
a(n) = (n-1)*a(n-1) + Sum_{i=1..n} binomial(n, i)*a(i-1)*a(n-i). - Vladimir Shevelev, Sep 30 2010
With an offset of 1, the e.g.f. is the compositional inverse ((x - 1)*log(1 - x))^(-1) = x + x^2/2! + 4*x^3/3! + 27*x^4/4! + .... - Peter Bala, Dec 09 2011
a(n) = denominator((1 + 1/n)^n) for n > 0. - Jean-François Alcover, Jan 14 2013
a(n) = A089072(n,n) for n > 0. - Reinhard Zumkeller, Mar 18 2013
a(n) = (n-1)^(n-1)*(2*n) + Sum_{i=1..n-2} binomial(n, i)*(i^i*(n-i-1)^(n-i-1)), n > 1, a(0) = 1, a(1) = 1. - Vladimir Kruchinin, Nov 28 2014
log(a(n)) = lim_{k->infinity} k*(n^(1+1/k) - n). - Richard R. Forberg, Feb 04 2015
From Ilya Gutkovskiy, Jun 18 2016: (Start)
Sum_{n>=1} 1/a(n) = 1.291285997... = A073009.
Sum_{n>=1} 1/a(n)^2 = 1.063887103... = A086648.
Sum_{n>=1} n!/a(n) = 1.879853862... = A094082. (End)
A000169(n+1)/a(n) -> e, as n -> oo. - Daniel Suteu, Jul 23 2016
a(n) = n!*Product_{k=1..n} binomial(n, k)/Product_{k=1..n-1} binomial(n-1, k) = n!*A001142(n)/A001142(n-1). - Tony Foster III, Sep 05 2018
a(n-1) = abs(p_n(2-n)), for n > 2, the single local extremum of the n-th row polynomial of A055137 with Bagula's sign convention. - Tom Copeland, Nov 15 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = A083648. - Amiram Eldar, Jun 25 2021
Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = e (see Brothers/Knox link). - Harlan J. Brothers, Oct 24 2021
Conjecture: a(n) = Sum_{i=0..n} A048994(n, i) * A048993(n+i, n) for n >= 0; proved by Mike Earnest, see link at A354797. - Werner Schulte, Jun 19 2022

A039599 Triangle formed from even-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 5, 1, 14, 28, 20, 7, 1, 42, 90, 75, 35, 9, 1, 132, 297, 275, 154, 54, 11, 1, 429, 1001, 1001, 637, 273, 77, 13, 1, 1430, 3432, 3640, 2548, 1260, 440, 104, 15, 1, 4862, 11934, 13260, 9996, 5508, 2244, 663, 135, 17, 1

Offset: 0

N. J. A. Sloane

T(n,k) is the number of lattice paths from (0,0) to (n,n) with steps E = (1,0) and N = (0,1) which touch but do not cross the line x - y = k and only situated above this line; example: T(3,2) = 5 because we have EENNNE, EENNEN, EENENN, ENEENN, NEEENN. - Philippe Deléham, May 23 2005
The matrix inverse of this triangle is the triangular matrix T(n,k) = (-1)^(n+k)* A085478(n,k). - Philippe Deléham, May 26 2005
Essentially the same as A050155 except with a leading diagonal A000108 (Catalan numbers) 1, 1, 2, 5, 14, 42, 132, 429, .... - Philippe Deléham, May 31 2005
Number of Grand Dyck paths of semilength n and having k downward returns to the x-axis. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)). Example: T(3,2)=5 because we have u(d)uud(d),uud(d)u(d),u(d)u(d)du,u(d)duu(d) and duu(d)u(d) (the downward returns to the x-axis are shown between parentheses). - Emeric Deutsch, May 06 2006
Riordan array (c(x),x*c(x)^2) where c(x) is the g.f. of A000108; inverse array is (1/(1+x),x/(1+x)^2). - Philippe Deléham, Feb 12 2007
The triangle may also be generated from M^n*[1,0,0,0,0,0,0,0,...], where M is the infinite tridiagonal matrix with all 1's in the super and subdiagonals and [1,2,2,2,2,2,2,...] in the main diagonal. - Philippe Deléham, Feb 26 2007
Inverse binomial matrix applied to A124733. Binomial matrix applied to A089942. - Philippe Deléham, Feb 26 2007
Number of standard tableaux of shape (n+k,n-k). - Philippe Deléham, Mar 22 2007
From Philippe Deléham, Mar 30 2007: (Start)
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1. Other triangles arise by choosing different values for (x,y):
(0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970
(1,0) -> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877;
(1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598;
(2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954;
(3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791;
(4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. (End)
The table U(n,k) = Sum_{j=0..n} T(n,j)*k^j is given in A098474. - Philippe Deléham, Mar 29 2007
Sequence read mod 2 gives A127872. - Philippe Deléham, Apr 12 2007
Number of 2n step walks from (0,0) to (2n,2k) and consisting of step u=(1,1) and d=(1,-1) and the path stays in the nonnegative quadrant. Example: T(3,0)=5 because we have uuuddd, uududd, ududud, uduudd, uuddud; T(3,1)=9 because we have uuuudd, uuuddu, uuudud, ududuu, uuduud, uduudu, uudduu, uduuud, uududu; T(3,2)=5 because we have uuuuud, uuuudu, uuuduu, uuduuu, uduuuu; T(3,3)=1 because we have uuuuuu. - Philippe Deléham, Apr 16 2007, Apr 17 2007, Apr 18 2007
Triangular matrix, read by rows, equal to the matrix inverse of triangle A129818. - Philippe Deléham, Jun 19 2007
Let Sum_{n>=0} a(n)*x^n = (1+x)/(1-mx+x^2) = o.g.f. of A_m, then Sum_{k=0..n} T(n,k)*a(k) = (m+2)^n. Related expansions of A_m are: A099493, A033999, A057078, A057077, A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, A097783, A077416, A126866, A028230, A161591, for m=-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, respectively. - Philippe Deléham, Nov 16 2009
The Kn11, Kn12, Fi1 and Fi2 triangle sums link the triangle given above with three sequences; see the crossrefs. For the definitions of these triangle sums, see A180662. - Johannes W. Meijer, Apr 20 2011
4^n = (n-th row terms) dot (first n+1 odd integer terms). Example: 4^4 = 256 = (14, 28, 20, 7, 1) dot (1, 3, 5, 7, 9) = (14 + 84 + 100 + 49 + 9) = 256. - Gary W. Adamson, Jun 13 2011
The linear system of n equations with coefficients defined by the first n rows solve for diagonal lengths of regular polygons with N= 2n+1 edges; the constants c^0, c^1, c^2, ... are on the right hand side, where c = 2 + 2*cos(2*Pi/N). Example: take the first 4 rows relating to the 9-gon (nonagon), N = 2*4 + 1; with c = 2 + 2*cos(2*Pi/9) = 3.5320888.... The equations are (1,0,0,0) = 1; (1,1,0,0) = c; (2,3,1,0) = c^2; (5,9,5,1) = c^3. The solutions are 1, 2.53208..., 2.87938..., and 1.87938...; the four distinct diagonal lengths of the 9-gon (nonagon) with edge = 1. (Cf. comment in A089942 which uses the analogous operations but with c = 1 + 2*cos(2*Pi/9).) - Gary W. Adamson, Sep 21 2011
Also called the Lobb numbers, after Andrew Lobb, are a natural generalization of the Catalan numbers, given by L(m,n)=(2m+1)*Binomial(2n,m+n)/(m+n+1), where n >= m >= 0. For m=0, we get the n-th Catalan number. See added reference. - Jayanta Basu, Apr 30 2013
From Wolfdieter Lang, Sep 20 2013: (Start)
T(n, k) = A053121(2*n, 2*k). T(n, k) appears in the formula for the (2*n)-th power of the algebraic number rho(N):= 2*cos(Pi/N) = R(N, 2) in terms of the odd-indexed diagonal/side length ratios R(N, 2*k+1) = S(2*k, rho(N)) in the regular N-gon inscribed in the unit circle (length unit 1). S(n, x) are Chebyshev's S polynomials (see A049310):
  rho(N)^(2*n) = Sum_{k=0..n} T(n, k)*R(N, 2*k+1), n >= 0, identical in N > = 1. For a proof see the Sep 21 2013 comment under A053121. Note that this is the unreduced version if R(N, j) with j > delta(N), the degree of the algebraic number rho(N) (see A055034), appears.
  For the odd powers of rho(n) see A039598. (End)
Unsigned coefficients of polynomial numerators of Eqn. 2.1 of the Chakravarty and Kodama paper, defining the polynomials of A067311. - Tom Copeland, May 26 2016
The triangle is the Riordan square of the Catalan numbers in the sense of A321620. - Peter Luschny, Feb 14 2023

			Triangle T(n, k) begins:
  n\k     0     1     2     3     4     5    6   7   8  9
  0:      1
  1:      1     1
  2:      2     3     1
  3:      5     9     5     1
  4:     14    28    20     7     1
  5:     42    90    75    35     9     1
  6:    132   297   275   154    54    11    1
  7:    429  1001  1001   637   273    77   13   1
  8:   1430  3432  3640  2548  1260   440  104  15   1
  9:   4862 11934 13260  9996  5508  2244  663 135  17  1
  ... Reformatted by _Wolfdieter Lang_, Dec 21 2015
From _Paul Barry_, Feb 17 2011: (Start)
Production matrix begins
  1, 1,
  1, 2, 1,
  0, 1, 2, 1,
  0, 0, 1, 2, 1,
  0, 0, 0, 1, 2, 1,
  0, 0, 0, 0, 1, 2, 1,
  0, 0, 0, 0, 0, 1, 2, 1 (End)
From _Wolfdieter Lang_, Sep 20 2013: (Start)
Example for rho(N) = 2*cos(Pi/N) powers:
n=2: rho(N)^4 = 2*R(N,1) + 3*R(N,3) + 1*R(N, 5) =
  2 + 3*S(2, rho(N)) + 1*S(4, rho(N)), identical in N >= 1. For N=4 (the square with only one distinct diagonal), the degree delta(4) = 2, hence R(4, 3) and R(4, 5) can be reduced, namely to R(4, 1) = 1 and R(4, 5) = -R(4,1) = -1, respectively. Therefore, rho(4)^4 =(2*cos(Pi/4))^4 = 2 + 3 -1 = 4. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
T. Myers and L. Shapiro, Some applications of the sequence 1, 5, 22, 93, 386, ... to Dyck paths and ordered trees, Congressus Numerant., 204 (2010), 93-104.

T. D. Noe, Rows n=0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Quang T. Bach and Jeffrey B. Remmel, Generating functions for descents over permutations which avoid sets of consecutive patterns, arXiv:1510.04319 [math.CO], 2015 (see p.25).
M. Barnabei, F. Bonetti and M. Silimbani, Two permutation classes enumerated by the central binomial coefficients, arXiv preprint arXiv:1301.1790 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.3.8
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010), Article 10.8.2, example 15.
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
Paul Barry, Comparing two matrices of generalized moments defined by continued fraction expansions, arXiv preprint arXiv:1311.7161 [math.CO], 2013 and J. Int. Seq. 17 (2014) # 14.5.1.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 15, 29.
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 12.
Jonathan E. Beagley and Paul Drube, Combinatorics of Tableau Inversions, Electron. J. Combin., 22 (2015), #P2.44.
S. Chakravarty and Y. Kodama, A generating function for the N-soliton solutions of the Kadomtsev-Petviashvili II equation, arXiv preprint arXiv:0802.0524v2 [nlin.SI], 2008.
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., Vol. 21 (2018), #18.2.1.
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 11.
Paul Drube, Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers, arXiv:1606.04869 [math.CO], 2016.
Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8.
T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, example p 32.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Thomas Koshy, Lobb's generalization of Catalan's parenthesization problem, The College Mathematics Journal 40 (2), March 2009, 99-107, DOI:10.1080/07468342.2009.11922344.
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 11.
Andrew Lobb, Deriving the n-th Catalan number, Mathematical Gazette, Vol. 83, No. 496 (March 1999), 109-110.
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174. See page 161.
Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016 (see 2.8).
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14, No. 4 (1957), 405-414: 124. [Note: there is a typo]
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp. 29 (129) (1975) 215-222
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths. Eur. J. Comb. 39, 157-169 (2014), Table 2.2.
Wikipedia, Lobb number
W.-J. Woan, L. Shapiro and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, 104 (1997), 926-931.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017), 3081-3091.

Columns give: A000108, A000245, A000344, A000588, A001392, A000589, A000590, A000012, A005408, A014107(n>1).
Row sums: A000984.
Triangle sums (see the comments): A000958 (Kn11), A001558 (Kn12), A088218 (Fi1, Fi2).
Cf. A008313, A039598, A084938, A000007, A067311, A321620.

/* As triangle */ [[Binomial(2*n, k+n)*(2*k+1)/(k+n+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 16 2015

T:=(n,k)->(2*k+1)*binomial(2*n,n-k)/(n+k+1): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form # Emeric Deutsch, May 06 2006
T := proc(n, k) option remember; if k = n then 1 elif k > n then 0 elif k = 0 then T(n-1, 0) + T(n-1,1) else T(n-1, k-1) + 2*T(n-1, k) + T(n-1, k+1) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..9) od; # Peter Luschny, Feb 14 2023

Table[Abs[Differences[Table[Binomial[2 n, n + i], {i, 0, n + 1}]]], {n, 0,7}] // Flatten (* Geoffrey Critzer, Dec 18 2011 *)
Join[{1},Flatten[Table[Binomial[2n-1,n-k]-Binomial[2n-1,n-k-2],{n,10},{k,0,n}]]] (* Harvey P. Dale, Dec 18 2011 *)
Flatten[Table[Binomial[2*n,m+n]*(2*m+1)/(m+n+1),{n,0,9},{m,0,n}]] (* Jayanta Basu, Apr 30 2013 *)

a(n, k) = (2*n+1)/(n+k+1)*binomial(2*k, n+k)
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(a(y, x), ", ")); print(""))
trianglerows(10) \\ Felix Fröhlich, Jun 24 2016

# Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle
def A039599_triangle(n) :
    D = [0]*(n+2); D[1] = 1
    b = True ; h = 1
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        if b : print([D[z] for z in (1..h-1)])
        b = not b
A039599_triangle(10)  # Peter Luschny, May 01 2012

T(n,k) = C(2*n-1, n-k) - C(2*n-1, n-k-2), n >= 1, T(0,0) = 1.
From Emeric Deutsch, May 06 2006: (Start)
T(n,k) = (2*k+1)*binomial(2*n,n-k)/(n+k+1).
G.f.: G(t,z)=1/(1-(1+t)*z*C), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function. (End)
The following formulas were added by Philippe Deléham during 2003 to 2009: (Start)
Triangle T(n, k) read by rows; given by A000012 DELTA A000007, where DELTA is Deléham's operator defined in A084938.
T(n, k) = C(2*n, n-k)*(2*k+1)/(n+k+1). Sum(k>=0; T(n, k)*T(m, k) = A000108(n+m)); A000108: numbers of Catalan.
T(n, 0) = A000108(n); T(n, k) = 0 if k>n; for k>0, T(n, k) = Sum_{j=1..n} T(n-j, k-1)*A000108(j).
T(n, k) = A009766(n+k, n-k) = A033184(n+k+1, 2k+1).
G.f. for column k: Sum_{n>=0} T(n, k)*x^n = x^k*C(x)^(2*k+1) where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108.
T(0, 0) = 1, T(n, k) = 0 if n<0 or n=1, T(n, k) = T(n-1, k-1) + 2*T(n-1, k) + T(n-1, k+1).
a(n) + a(n+1) = 1 + A000108(m+1) if n = m*(m+3)/2; a(n) + a(n+1) = A039598(n) otherwise.
T(n, k) = A050165(n, n-k).
Sum_{j>=0} T(n-k, j)*A039598(k, j) = A028364(n, k).
Matrix inverse of the triangle T(n, k) = (-1)^(n+k)*binomial(n+k, 2*k) = (-1)^(n+k)*A085478(n, k).
Sum_{k=0..n} T(n, k)*x^k = A000108(n), A000984(n), A007854(n), A076035(n), A076036(n) for x = 0, 1, 2, 3, 4.
Sum_{k=0..n} (2*k+1)*T(n, k) = 4^n.
T(n, k)*(-2)^(n-k) = A114193(n, k).
Sum_{k>=h} T(n,k) = binomial(2n,n-h).
Sum_{k=0..n} T(n,k)*5^k = A127628(n).
Sum_{k=0..n} T(n,k)*7^k = A115970(n).
T(n,k) = Sum_{j=0..n-k} A106566(n+k,2*k+j).
Sum_{k=0..n} T(n,k)*6^k = A126694(n).
Sum_{k=0..n} T(n,k)*A000108(k) = A007852(n+1).
Sum_{k=0..floor(n/2)} T(n-k,k) = A000958(n+1).
Sum_{k=0..n} T(n,k)*(-1)^k = A000007(n).
Sum_{k=0..n} T(n,k)*(-2)^k = (-1)^n*A064310(n).
T(2*n,n) = A126596(n).
Sum_{k=0..n} T(n,k)*(-x)^k = A000007(n), A126983(n), A126984(n), A126982(n), A126986(n), A126987(n), A127017(n), A127016(n), A126985(n), A127053(n) for x=1,2,3,4,5,6,7,8,9,10 respectively.
Sum_{j>=0} T(n,j)*binomial(j,k) = A116395(n,k).
T(n,k) = Sum_{j>=0} A106566(n,j)*binomial(j,k).
T(n,k) = Sum_{j>=0} A127543(n,j)*A038207(j,k).
Sum_{k=0..floor(n/2)} T(n-k,k)*A000108(k) = A101490(n+1).
T(n,k) = A053121(2*n,2*k).
Sum_{k=0..n} T(n,k)*sin((2*k+1)*x) = sin(x)*(2*cos(x))^(2*n).
T(n,n-k) = Sum_{j>=0} (-1)^(n-j)*A094385(n,j)*binomial(j,k).
Sum_{j>=0} A110506(n,j)*binomial(j,k) = Sum_{j>=0} A110510(n,j)*A038207(j,k) = T(n,k)*2^(n-k).
Sum_{j>=0} A110518(n,j)*A027465(j,k) = Sum_{j>=0} A110519(n,j)*A038207(j,k) = T(n,k)*3^(n-k).
Sum_{k=0..n} T(n,k)*A001045(k) = A049027(n), for n>=1.
Sum_{k=0..n} T(n,k)*a(k) = (m+2)^n if Sum_{k>=0} a(k)*x^k = (1+x)/(x^2-m*x+1).
Sum_{k=0..n} T(n,k)*A040000(k) = A001700(n).
Sum_{k=0..n} T(n,k)*A122553(k) = A051924(n+1).
Sum_{k=0..n} T(n,k)*A123932(k) = A051944(n).
Sum_{k=0..n} T(n,k)*k^2 = A000531(n), for n>=1.
Sum_{k=0..n} T(n,k)*A000217(k) = A002457(n-1), for n>=1.
Sum{j>=0} binomial(n,j)*T(j,k)= A124733(n,k).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
Sum_{k=0..n} T(n,k)*A005043(k) = A127632(n).
Sum_{k=0..n} T(n,k)*A132262(k) = A089022(n).
T(n,k) + T(n,k+1) = A039598(n,k).
T(n,k) = A128899(n,k)+A128899(n,k+1).
Sum_{k=0..n} T(n,k)*A015518(k) = A076025(n), for n>=1. Also Sum_{k=0..n} T(n,k)*A015521(k) = A076026(n), for n>=1.
Sum_{k=0..n} T(n,k)*(-1)^k*x^(n-k) = A033999(n), A000007(n), A064062(n), A110520(n), A132863(n), A132864(n), A132865(n), A132866(n), A132867(n), A132869(n), A132897(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*(-1)^(k+1)*A000045(k) = A109262(n), A000045:= Fibonacci numbers.
Sum_{k=0..n} T(n,k)*A000035(k)*A016116(k) = A143464(n).
Sum_{k=0..n} T(n,k)*A016116(k) = A101850(n).
Sum_{k=0..n} T(n,k)*A010684(k) = A100320(n).
Sum_{k=0..n} T(n,k)*A000034(k) = A029651(n).
Sum_{k=0..n} T(n,k)*A010686(k) = A144706(n).
Sum_{k=0..n} T(n,k)*A006130(k-1) = A143646(n), with A006130(-1)=0.
T(n,2*k)+T(n,2*k+1) = A118919(n,k).
Sum_{k=0..j} T(n,k) = A050157(n,j).
Sum_{k=0..2} T(n,k) = A026012(n); Sum_{k=0..3} T(n,k)=A026029(n).
Sum_{k=0..n} T(n,k)*A000045(k+2) = A026671(n).
Sum_{k=0..n} T(n,k)*A000045(k+1) = A026726(n).
Sum_{k=0..n} T(n,k)*A057078(k) = A000012(n).
Sum_{k=0..n} T(n,k)*A108411(k) = A155084(n).
Sum_{k=0..n} T(n,k)*A057077(k) = 2^n = A000079(n).
Sum_{k=0..n} T(n,k)*A057079(k) = 3^n = A000244(n).
Sum_{k=0..n} T(n,k)*(-1)^k*A011782(k) = A000957(n+1).
(End)
T(n,k) = Sum_{j=0..k} binomial(k+j,2j)*(-1)^(k-j)*A000108(n+j). - Paul Barry, Feb 17 2011
Sum_{k=0..n} T(n,k)*A071679(k+1) = A026674(n+1). - Philippe Deléham, Feb 01 2014
Sum_{k=0..n} T(n,k)*(2*k+1)^2 = (4*n+1)*binomial(2*n,n). - Werner Schulte, Jul 22 2015
Sum_{k=0..n} T(n,k)*(2*k+1)^3 = (6*n+1)*4^n. - Werner Schulte, Jul 22 2015
Sum_{k=0..n} (-1)^k*T(n,k)*(2*k+1)^(2*m) = 0 for 0 <= m < n (see also A160562). - Werner Schulte, Dec 03 2015
T(n,k) = GegenbauerC(n-k,-n+1,-1) - GegenbauerC(n-k-1,-n+1,-1). - Peter Luschny, May 13 2016
T(n,n-2) = A014107(n). - R. J. Mathar, Jan 30 2019
T(n,n-3) = n*(2*n-1)*(2*n-5)/3. - R. J. Mathar, Jan 30 2019
T(n,n-4) = n*(n-1)*(2*n-1)*(2*n-7)/6. - R. J. Mathar, Jan 30 2019
T(n,n-5) = n*(n-1)*(2*n-1)*(2*n-3)*(2*n-9)/30. - R. J. Mathar, Jan 30 2019

Corrected by Philippe Deléham, Nov 26 2009, Dec 14 2009

A006950 G.f.: Product_{k>=1} (1 + x^(2k - 1)) / (1 - x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 196, 236, 287, 350, 420, 501, 602, 722, 858, 1016, 1206, 1431, 1687, 1981, 2331, 2741, 3206, 3740, 4368, 5096, 5922, 6868, 7967, 9233, 10670, 12306, 14193, 16357, 18803, 21581

Offset: 0

N. J. A. Sloane, Warren D. Smith

nonn

Also the number of partitions of n in which all odd parts are distinct. There is no restriction on the even parts. E.g., a(9)=13 because "9 = 8+1 = 7+2 = 6+3 = 6+2+1 = 5+4 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 4+2+2+1 = 3+2+2+2 = 2+2+2+2+1". - Noureddine Chair, Feb 03 2005
Number of partitions of n in which each even part occurs with even multiplicity. There is no restriction on the odd parts.
Also the number of partitions of n into parts not congruent to 2 mod 4. - James Sellers, Feb 08 2002
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras o(n) of skew-symmetric n X n matrices, n=0,1,2,3,... (the cases n=0,1 being degenerate). This sequence, A015128 and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=2.
Equals polcoeff inverse of A010054 with alternate signs. - Gary W. Adamson, Mar 15 2010
It appears that this sequence is related to the generalized hexagonal numbers (A000217) in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: this is 1 together with the row sums of triangle A195836, also column 1 of A195836, also column 2 of the square array A195825. - Omar E. Pol, Oct 09 2011
Since this is also column 2 of A195825 so the sequence contains only one plateau [1, 1, 1] of level 1 and length 3. For more information see A210843. - Omar E. Pol, Jun 27 2012
Convolution of A035363 and A000700. - Vaclav Kotesovec, Aug 17 2015
Also the number of ways to stack n triangles in a valley (pointing upwards or downwards depending on row parity). - Seiichi Manyama, Jul 07 2018

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...
G.f. = q^-1 + q^7 + q^15 + 2*q^23 + 3*q^31 + 4*q^39 + 5*q^47 + 7*q^55 + 10*q^63 + ...
From _Seiichi Manyama_, Jul 07 2018: (Start)
n | the ways to stack n triangles in a valley
--+------------------------------------------------------
1 | *---*
  |  \ /
  |   *
  |
2 |   *
  |  / \
  | *---*
  |  \ /
  |   *
  |
3 |   *---*     *---*
  |  / \ /       \ / \
  | *---*         *---*
  |  \ /           \ /
  |   *             *
  |
4 |     *                       *
  |    / \                     / \
  |   *---*     *---*---*     *---*
  |  / \ /       \ / \ /       \ / \
  | *---*         *---*         *---*
  |  \ /           \ /           \ /
  |   *             *             *
  |
5 |     *---*         *         *         *---*
  |    / \ /         / \       / \         \ / \
  |   *---*     *---*---*     *---*---*     *---*
  |  / \ /       \ / \ /       \ / \ /       \ / \
  | *---*         *---*         *---*         *---*
  |  \ /           \ /           \ /           \ /
  |   *             *             *             *
  |
6 |       *
  |      / \
  |     *---*         *---*     *   *     *---*
  |    / \ /         / \ /     / \ / \     \ / \
  |   *---*     *---*---*     *---*---*     *---*---*
  |  / \ /       \ / \ /       \ / \ /       \ / \ /
  | *---*         *---*         *---*         *---*
  |  \ /           \ /           \ /           \ /
  |   *             *             *             *
  |   *
  |  / \
  | *---*
  |  \ / \
  |   *---*
  |    \ / \
  |     *---*
  |      \ /
  |       *
(End)

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.
M. D. Hirschhorn, The Power of q, Springer, 2017. See pod, page 297.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Luca Ferrari, Schröder partitions, Schröder tableaux and weak poset patterns, arXiv:1606.06624 [math.CO], 2016. Mentions this sequence.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See Example 4.2 p. 13.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
L. Wang, New Congruences for Partitions where the Odd Parts are Distinct, J. Int. Seq. 18 (2015) # 15.4.2.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012

See also Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cf. A015128, A046682, A106459.
Cf. A163203.
Cf. A010054, A085642, A316384.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 06 2013

CoefficientList[ Series[ Product[(1 + x^(2k - 1))/(1 - x^(2k)), {k, 25}], {x, 0, 50}], x] (* Robert G. Wilson v, Jun 28 2012 *)
CoefficientList[Series[x*QPochhammer[-1/x, x^2] / ((1+x)*QPochhammer[x^2, x^2]), {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
CoefficientList[Series[2*(-x)^(1/8) / EllipticTheta[2, 0, Sqrt[-x]], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-Mod[i, 2]]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz *)

{a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, (-1)^(m-d)*d)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Jul 22 2009
(GW-BASIC)
' A program with two A-numbers (Note that here A000217 are the generalized hexagonal numbers):
10 Dim A000217(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 51: For j = 1 to n
30 If A000217(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A000217(j))
40 Next j: Print a(n-1);: Next n ' Omar E. Pol, Jun 10 2012

a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*A002129(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Feb 05 2002
G.f.: 1/Sum_{k>=0} (-x)^(k*(k+1)/2). - Vladeta Jovovic, Sep 22 2002 [corrected by Vaclav Kotesovec, Aug 17 2015]
a(n) = A059777(n-1)+A059777(n), n > 0. - Vladeta Jovovic, Sep 22 2002
G.f.: Product_{m>=1} (1+x^m)^(if A001511(m) > 1, A001511(m)-1 else A001511(m)). - Jon Perry, Apr 15 2005
Expansion of 1 / psi(-x) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(1/8) * eta(q^2) / (eta(q) * eta(q^4)) in powers of q.
Convolution inverse of A106459. - Michael Somos, Nov 02 2005
G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d] * x^n/n ). - Paul D. Hanna, Jul 22 2009
a(n) ~ (8*n+1) * cosh(sqrt(8*n-1)*Pi/4) / (16*sqrt(2)*n^2) - sinh(sqrt(8*n-1)*Pi/4) / (2*Pi*n^(3/2)) ~ exp(Pi*sqrt(n/2))/(4*sqrt(2)*n) * (1 - (2/Pi + Pi/16)/sqrt(2*n) + (3/16 + Pi^2/1024)/n). - Vaclav Kotesovec, Aug 17 2015, extended Jan 09 2017
Can be computed recursively by Sum_{j>=0} (-1)^(ceiling(j/2)) a(n - j(j+1)/2) = 0, for n > 0. [Merca, Theorem 4.3] - Eric M. Schmidt, Sep 21 2017
a(n) = A000041(n) - A085642(n), for n >= 1. - Wouter Meeussen, Dec 20 2017

G.f. and more terms from Vladeta Jovovic, Feb 05 2002

A009545 Expansion of e.g.f. sin(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32, 32, 0, -64, -128, -128, 0, 256, 512, 512, 0, -1024, -2048, -2048, 0, 4096, 8192, 8192, 0, -16384, -32768, -32768, 0, 65536, 131072, 131072, 0, -262144, -524288, -524288, 0, 1048576, 2097152, 2097152, 0, -4194304, -8388608, -8388608, 0, 16777216, 33554432

Offset: 0

R. H. Hardin

Also first of the two associated sequences a(n) and b(n) built from a(0)=0 and b(0)=1 with the formulas a(n) = a(n-1) + b(n-1) and b(n) = -a(n-1) + b(n-1). The initial terms of the second sequence b(n) are 1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32, -64, -64, 0, 128, 256, ... The points Mn(a(n)+b(n)*I) of the complex plane are located on the spiral logarithmic rho = 2*(1/2)^(2*theta)/Pi) and on the straight lines drawn from the origin with slopes: infinity, 1/2, 0, -1/2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007
A000225: (1, 3, 7, 15, 31, ...) = 2^n - 1 = INVERT transform of A009545 starting (1, 2, 2, 0, -4, -8, ...). (Cf. comments in A144081). - Gary W. Adamson, Sep 10 2008
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012
The variant 0, 1, -2, 2, 0, -4, 8, -8, 0, 16, -32, 32, 0, -64, (with different signs) is the Lucas U(-2,2) sequence. - R. J. Mathar, Jan 08 2013
(1+i)^n = A146559(n) + a(n)*i where i = sqrt(-1). - Philippe Deléham, Feb 13 2013
This is the Lucas U(2,2) sequence. - Raphie Frank, Nov 28 2015
{A146559, A009545} are the difference analogs of {cos(x),sin(x)} (cf. [Shevelev] link). - Vladimir Shevelev, Jun 08 2017

N. J. A. Sloane, Table of n, a(n) for n = 0..2000, Apr 09 2016 (first 100 terms from T. D. Noe)
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=2, q=-2.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=2.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
N. J. A. Sloane, Table of n, (I-1)^n for n = 0..100
Wikipedia, Lucas sequence.
Index entries for linear recurrences with constant coefficients, signature (2,-2).
Index entries for sequences related to Chebyshev polynomials
Index entries for Lucas sequences

Cf. A009116. For minor variants of this sequence see A108520, A084102, A099087.
a(2*n) = A056594(n)*2^n, n >= 1, a(2*n+1) = A057077(n)*2^n.
This is the next term in the sequence A015518, A002605, A000129, A000079, A001477.
Cf. A000225, A144081. - Gary W. Adamson, Sep 10 2008
Cf. A146559.

I:=[0,1,2,2]; [n le 4 select I[n] else -4*Self(n-4): n in [1..60]]; // Vincenzo Librandi, Nov 29 2015

t1 := sum(n*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 0 to 50 do printf(`%d, `, coeff(F, x, i)) od: # Zerinvary Lajos, Mar 22 2009
G(x):=exp(x)*sin(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..50 ); # Zerinvary Lajos, Apr 05 2009
A009545 := n -> `if`(n<2, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], 2)):
seq(simplify(A009545(n)), n=0..50); # Peter Luschny, Dec 17 2015

nn=104; Range[0,nn-1]! CoefficientList[Series[Sin[x]Exp[x], {x,0,nn}], x] (* T. D. Noe, May 26 2007 *)
Join[{a=0,b=1},Table[c=2*b-2*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
f[n_] := (1 + I)^(n - 2) + (1 - I)^(n - 2); Array[f, 51, 0] (* Robert G. Wilson v, May 30 2011 *)
LinearRecurrence[{2,-2},{0,1},110] (* Harvey P. Dale, Oct 13 2011 *)

x='x+O('x^66); Vec(serlaplace(exp(x)*sin(x))) /* Joerg Arndt, Apr 24 2011 */

x='x+O('x^100); concat(0, Vec(x/(1-2*x+2*x^2))) \\ Altug Alkan, Dec 04 2015

def A009545(n): return ((0, 1, 2, 2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1) # Chai Wah Wu, Feb 16 2024

[lucas_number1(n,2,2) for n in range(0, 51)] # Zerinvary Lajos, Apr 23 2009

def A146559():
    x, y = 0, -1
    while True:
        yield x
        x, y = x - y, x + y
a = A146559(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

a(0)=0; a(1)=1; a(2)=2; a(3)=2; a(n) = -4*a(n-4), n>3. - Larry Reeves (larryr(AT)acm.org), Aug 24 2000
Imaginary part of (1+i)^n. - Marc LeBrun
G.f.: x/(1 - 2*x + 2*x^2).
E.g.f.: sin(x)*exp(x).
a(n) = S(n-1, sqrt(2))*(sqrt(2))^(n-1) with S(n, x)= U(n, x/2) Chebyshev's polynomials of the 2nd kind, Cf. A049310, S(-1, x) := 0.
a(n) = ((1+i)^n - (1-i)^n)/(2*i) = 2*a(n-1) - 2*a(n-2) (with a(0)=0 and a(1)=1). - Henry Bottomley, May 10 2001
a(n) = (1+i)^(n-2) + (1-i)^(n-2). - Benoit Cloitre, Oct 28 2002
a(n) = Sum_{k=0..n-1} (-1)^floor(k/2)*binomial(n-1, k). - Benoit Cloitre, Jan 31 2003
a(n) = 2^(n/2)sin(Pi*n/4). - Paul Barry, Sep 17 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k+1)*(-1)^k. - Paul Barry, Sep 20 2003
a(n+1) = Sum_{k=0..n} 2^k*A109466(n,k). - Philippe Deléham, Nov 13 2006
a(n) = 2*((1/2)^(2*theta(n)/Pi))*cos(theta(n)) where theta(4*p+1) = p*Pi + Pi/2, theta(4*p+2) = p*Pi + Pi/4, theta(4*p+3) = p*Pi - Pi/4, theta(4*p+4) = p*Pi - Pi/2, or a(0)=0, a(1)=1, a(2)=2, a(3)=2, and for n>3 a(n)=-4*a(n-4). Same formulas for the second sequence replacing cosines with sines. For example: a(0) = 0, b(0) = 1; a(1) = 0+1 = 1, b(1) = -0+1 = 1; a(2) = 1+1 = 2, b(2) = -1+1 = 0; a(3) = 2+0 = 2, b(3) = -2+0 = -2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n > 3, which implies the sequence is identical to its fourth differences. Binomial transform of 0, 1, 0, -1. - Paul Curtz, Dec 21 2007
Logarithm g.f. arctan(x/(1-x)) = Sum_{n>0} a(n)/n*x^n. - Vladimir Kruchinin, Aug 11 2010
a(n) = A046978(n) * A016116(n). - Paul Curtz, Apr 24 2011
E.g.f.: exp(x) * sin(x) = x + x^2/(G(0)-x); G(k) = 2k + 1 + x - x*(2k+1)/(4k+3+x+x^2*(4k+3)/( (2k+2)*(4k+5) - x^2 - x*(2k+2)*(4k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011
a(n) = Im( (1+i)^n ) where i=sqrt(-1). - Stanislav Sykora, Jun 11 2012
G.f.: x*U(0) where U(k) = 1 + x*(k+3) - x*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
G.f.: G(0)*x/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
G.f.: x + x^2*W(0), where W(k) = 1 + 1/(1 - x*(k+1)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 2*x)/( x*(4*k+4 - 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 06 2013
a(n) = (A^n - B^n)/(A - B), where A = 1 + i and B = 1 - i; A and B are solutions of x^2 - 2*x + 2 = 0. - Raphie Frank, Nov 28 2015
a(n) = 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], 2) for n >= 2. - Peter Luschny, Dec 17 2015
a(k+m) = a(k)*A146559(m) + a(m)*A146559(k). - Vladimir Shevelev, Jun 08 2017

Extended with signs by Olivier Gérard, Mar 15 1997
More terms from Larry Reeves (larryr(AT)acm.org), Aug 24 2000
Definition corrected by Joerg Arndt, Apr 24 2011

A195825 Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 7, 3, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 15, 5, 3, 1, 1, 1, 1, 1, 22, 7, 4, 2, 1, 1, 1, 1, 1, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1, 56, 16, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 77, 21, 10, 4

Offset: 0

Omar E. Pol, Sep 24 2011

In the infinite square array the column k is related to the generalized m-gonal numbers, where m = k+4. For example: the first column is related to the generalized pentagonal numbers A001318. The second column is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on ... (see the program in which A195152 is a table of generalized m-gonal numbers).
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041 (see below the first row of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A195825
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.                                          with a(0)=1]
========================================================
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
Conjecture: if k is odd then column k contains (k+1)/2 plateaus whose levels are the first (k+1)/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 2. Otherwise, if k is even then column k contains k/2 plateaus whose levels are the first k/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 3. The sequence A210843 gives the levels of the plateaus of column k, when k -> infinity. For the visualization of the plateaus see the graph of a column, for example see the graph of A210964. - Omar E. Pol, Jun 21 2012

			Array begins:
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    2,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    3,  2,  1,  1,  1,  1,  1,  1,  1,  1, ...
    5,  3,  2,  1,  1,  1,  1,  1,  1,  1, ...
    7,  4,  3,  2,  1,  1,  1,  1,  1,  1, ...
   11,  5,  4,  3,  2,  1,  1,  1,  1,  1, ...
   15,  7,  4,  4,  3,  2,  1,  1,  1,  1, ...
   22, 10,  5,  4,  4,  3,  2,  1,  1,  1, ...
   30, 13,  7,  4,  4,  4,  3,  2,  1,  1, ...
   42, 16, 10,  5,  4,  4,  4,  3,  2,  1, ...
   56, 21, 12,  7,  4,  4,  4,  4,  3,  2, ...
   77, 28, 14, 10,  5,  4,  4,  4,  4,  3, ...
  101, 35, 16, 12,  7,  4,  4,  4,  4,  4, ...
  135, 43, 21, 13, 10,  5,  4,  4,  4,  4, ...
  176, 55, 27, 14, 12,  7,  4,  4,  4,  4, ...
  ...
Column 1 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11, ... The column contains only one plateau: [1, 1] which has level 1 and length 2.
Column 3 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10, ... The column contains two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2.
Column 6 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21, ... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3.

Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Columns (1-10): A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A211970.
Cf. A057077, A195152, A210843.

Column k is asymptotic to exp(Pi*sqrt(2*n/(k+2))) / (8*sin(Pi/(k+2))*n). - Vaclav Kotesovec, Aug 14 2017

A054320 Expansion of g.f.: (1 + x)/(1 - 10*x + x^2).

Original entry on oeis.org

1, 11, 109, 1079, 10681, 105731, 1046629, 10360559, 102558961, 1015229051, 10049731549, 99482086439, 984771132841, 9748229241971, 96497521286869, 955226983626719, 9455772314980321, 93602496166176491, 926569189346784589, 9172089397301669399, 90794324783669909401

Offset: 0

Ignacio Larrosa Cañestro, Feb 27 2000

Chebyshev's even-indexed U-polynomials evaluated at sqrt(3).
a(n)^2 is a star number (A003154).
Any k in the sequence has the successor 5*k + 2*sqrt(3(2*k^2 + 1)). - Lekraj Beedassy, Jul 08 2002
{a(n)} give the values of x solving: 3*y^2 - 2*x^2 = 1. Corresponding values of y are given by A072256(n+1). x + y = A001078(n+1). - Richard R. Forberg, Nov 21 2013
The aerated sequence (b(n))n>=1 = [1, 0, 11, 0, 109, 0, 1079, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -8, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - Peter Bala, Mar 22 2015

			a(1)^2 = 121 is the 5th star number (A003154).

G. C. Greubel, Table of n, a(n) for n = 0..1000
K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math. 3 (2) (2008), pp. 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Star Number
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (10,-1).
Index entries for sequences related to Chebyshev polynomials.

A member of the family A057078, A057077, A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, which are the expansions of (1+x) / (1-kx+x^2) with k = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - Philippe Deléham, May 04 2004
Cf. A003154, A006061, A031138, A007667, A004189.
Cf. A138281. Cf. A100047.
Cf. A142238.

a:=[1,11];; for n in [3..30] do a[n]:=10*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jul 22 2019

I:=[1,11]; [n le 2 select I[n] else 10*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015

CoefficientList[Series[(1+x)/(1-10x+x^2), {x,0,30}], x] (* Vincenzo Librandi, Mar 22 2015 *)
a[c_, n_] := Module[{},
   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Numerator[Convergents[Sqrt[c], n p]];
   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
   Return[t];
] (* Complement of A142238 *)
a[3/2, 20] (* Gerry Martens, Jun 07 2015 *)

a(n)=subst(poltchebi(n+1)-poltchebi(n),x,5)/4;

(a(n)-1)^2 + a(n)^2 + (a(n)+1)^2 = b(n)^2 + (b(n)+1)^2 = c(n), where b(n) is A031138 and c(n) is A007667.
a(n) = 10*a(n-1) - a(n-2).
a(n) = (sqrt(6) - 2)/4*(5 + 2*sqrt(6))^(n+1) - (sqrt(6) + 2)/4*(5 - 2*sqrt(6))^(n+1).
a(n) = U(2*(n-1), sqrt(3)) = S(n-1, 10) + S(n-2, 10) with Chebyshev's U(n, x) and S(n, x) := U(n, x/2) polynomials and S(-1, x) := 0. S(n, 10) = A004189(n+1), n >= 0.
6*a(n)^2 + 3 is a square. Limit_{n->oo} a(n)/a(n-1) = 5 + 2*sqrt(6). - Gregory V. Richardson, Oct 13 2002
Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i), then (-1)^n*q(n, -12) = a(n). - Benoit Cloitre, Nov 10 2002
a(n) = L(n,-10)*(-1)^n, where L is defined as in A108299; see also A072256 for L(n,+10). - Reinhard Zumkeller, Jun 01 2005
From Reinhard Zumkeller, Mar 12 2008: (Start)
(sqrt(2) + sqrt(3))^(2*n+1) = a(n)*sqrt(2) + A138288(n)*sqrt(3);
a(n) = A138288(n) + A001078(n).
a(n) = A001079(n) + 3*A001078(n). (End)
a(n) = A142238(2n) = A041006(2n)/2 = A041038(2n)/4. - M. F. Hasler, Feb 14 2009
a(n) = sqrt(A006061(n)). - Zak Seidov, Oct 22 2012
a(n) = sqrt((3*A072256(n)^2 - 1)/2). - T. D. Noe, Oct 23 2012
(sqrt(3) + sqrt(2))^(2*n+1) - (sqrt(3) - sqrt(2))^(2*n+1) = a(n)*sqrt(8). - Bruno Berselli, Oct 29 2019
a(n) = A004189(n)+A004189(n+1). - R. J. Mathar, Oct 01 2021
E.g.f.: exp(5*x)*(2*cosh(2*sqrt(6)*x) + sqrt(6)*sinh(2*sqrt(6)*x))/2. - Stefano Spezia, May 16 2023
From Peter Bala, May 09 2025: (Start)
a(n) = Dir(n, 5), where Dir(n, x) denotes the n-th row polynomial of the triangle A244419.
a(n)^2 - 10*a(n)*a(n+1) + a(n+1)^2 = 12.
More generally, for arbitrary x, a(n+x)^2 - 10*a(n+x)*a(n+x+1) + a(n+x+1)^2 = 12 with a(n) := (sqrt(6) - 2)/4*(5 + 2*sqrt(6))^(n+1) - (sqrt(6) + 2)/4*(5 - 2*sqrt(6))^(n+1) as given above.
a(n+1/2) = sqrt(3) * A001078(n+1).
a(n+3/4) + a(n+1/4) = sqrt(6)*sqrt(sqrt(3) + 1) * A001078(n+1).
a(n+3/4) - a(n+1/4) = sqrt(sqrt(3) - 1) * A001079(n+1).
Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/12 (telescoping series: for n >= 1, 1/(a(n) - 1/a(n)) = 1/A004291(n) + 1/A004291(n+1)).
Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(3/2) (telescoping product: Product_{n = 1..k} ((a(n) + 1)/(a(n) - 1))^2 = 3/2 * (1 - 1/A171640(k+2))). (End)

Chebyshev comments from Wolfdieter Lang, Oct 31 2002

A051159 Triangle read by rows: T(n, k) = binomial(n mod 2, k mod 2) * binomial(n div 2, k div 2), where 'div' denotes integer division.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 3, 0, 3, 0, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1, 1, 0, 6, 0, 15, 0, 20, 0, 15, 0, 6, 0, 1, 1, 1, 6, 6, 15, 15, 20, 20, 15, 15, 6, 6, 1, 1

Offset: 0

Michael Somos, Oct 14 1999

Previous name: Triangular array made of three copies of Pascal's triangle.
Computing each term modulo 2 also gives A047999, i.e., a(n) mod 2 = A007318(n) mod 2 for all n. (The triangle is paritywise isomorphic to Pascal's Triangle.) - Antti Karttunen
5th row/column gives entries of A000217 (triangular numbers C(n+1,2)) repeated twice and every other entry in 6th row/column form A000217. 7th row/column gives entries of A000292 (Tetrahedral (or pyramidal) nos: C(n+3,3)) repeated twice and every other entry in 8th row/column form A000292. 9th row/column gives entries of A000332 (binomial coefficients binomial(n,4)) repeated twice and every other entry in 10th row/column form A000332. 11th row/column gives entries of A000389 (binomial coefficients C(n,5)) repeated twice and every other entry in 12th row/column form A000389. - Gerald McGarvey, Aug 21 2004
If Sum_{k=0..n} A(k)*T(n,k) = B(n), the sequence B is the S-D transform of the sequence A. - Philippe Deléham, Aug 02 2006
Number of n-bead black-white reversible strings with k black beads; also binary grids; string is palindromic. - Yosu Yurramendi, Aug 07 2008
Row sums give A016116(n+1). - Yosu Yurramendi, Aug 07 2008 [corrected by Petros Hadjicostas, Nov 04 2017]
Coefficients in expansion of (x + y)^n where x and y anticommute (y x = -x y), that is, q-binomial coefficients when q = -1. - Michael Somos, Feb 16 2009
The sequence of coefficients of a general polynomial recursion that links at w=2 to the Pascal triangle is here w=0. Row sums are {1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, ...}. - Roger L. Bagula and Gary W. Adamson, Dec 04 2009
T(n,k) is the number of palindromic compositions of n+1 with exactly k+1 parts. T(6,4) = 3 because we have the following compositions of n+1=7 with length k+1=5: 1+1+3+1+1, 2+1+1+1+2, 1+2+1+2+1. - Geoffrey Critzer, Mar 15 2014 [corrected by Petros Hadjicostas, Nov 03 2017]
Let P(n,k) be the number of palindromic compositions of n with exactly k parts. MacMahon (1893) was the first to prove that P(n,k) = T(n-1,k-1), where T(n,k) are the numbers in this sequence (see the comment above by G. Critzer). He actually proved that, for 1 <= s <= m, we have P(2*m,2*s) = P(2*m,2*s-1) = P(2*m-1, 2*s-1) = bin(m-1, s-1), but  P(2*m-1, 2*s) = 0. For the current sequence, this can be translated into T(2*m-1, 2*s-1) = T(2*m-1,2*s-2) = T(2*m-2, 2*s-2) = bin(m-1,s-1), but T(2m-2, 2*s-1) = 0 (valid again for 1 <= s <= m). - Petros Hadjicostas, Nov 03 2017
T is the infinite lower triangular matrix for this sequence; define two others, U and V; let U(n,k)=e_k(-1,2,-3,...,(-1)^n n), where e_k is the k-th elementary symmetric polynomial, and let V be the diagonal matrix A057077 (periodic sequence 1,1,-1,-1). Clearly V^-1 = V. Conjecture: U = U^-1, T = U . V, T^-1 = V . U, and |T| = |U|. - George Beck, Dec 16 2017
Let T*(n,k)=T(n,k) except when n is odd and k=(n+1)/2, where T*(n,k) = T(n,k)+2^((n-1)/2). Thus, T*(n,k) is the number of non-isomorphic symmetric stairs with n cells and k steps, i.e., k-1 changes of direction. See A016116. - Christian Barrientos and Sarah Minion, Jul 29 2018

			Triangle starts:
{1},
{1,  1},
{1,  0,  1},
{1,  1,  1,  1},
{1,  0,  2,  0,  1},
{1,  1,  2,  2,  1,  1},
{1,  0,  3,  0,  3,  0,  1},
{1,  1,  3,  3,  3,  3,  1,  1},
{1,  0,  4,  0,  6,  0,  4,  0,  1},
{1,  1,  4,  4,  6,  6,  4,  4,  1,  1},
{1,  0,  5,  0, 10,  0, 10,  0,  5,  0,  1},
{1,  1,  5,  5, 10, 10, 10, 10,  5,  5,  1,  1}
... - _Roger L. Bagula_ and _Gary W. Adamson_, Dec 04 2009

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Nantel Bergeron, Kelvin Chan, Yohana Solomon, Farhad Soltani, and Mike Zabrocki, Quasisymmetric harmonics of the exterior algebra, arXiv:2206.02065 [math.CO], 2022.
E. Burlachenko, Fractal generalized Pascal matrices, arXiv:1612.00970 [math.NT], 2016. See p. 3.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774. See Table 1 on p. 763.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).
M. E. Horn, The Didactical Relevance of the Pauli Pascal Triangle, arXiv:physics/0611277 [physics.ed-ph], 2006. [_Michael Somos_]
F. Al-Kharousi, R. Kehinde, and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Volume 58 (3) (2014), 363-375.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A051160, A016116, A034851, A169623, A036355.

a051159 n k = a051159_tabl !! n !! k
a051159_row n = a051159_tabl !! n
a051159_tabl = [1] : f [1] [1,1] where
   f us vs = vs : f vs (zipWith (+) ([0,0] ++ us) (us ++ [0,0]))
-- Reinhard Zumkeller, Apr 25 2013

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
      `if`(n<0 or k<0, 0, `if`(irem(n, 2)=1 or
       irem(k, 2)=0, T(n-1, k-1) + T(n-1, k), 0)))
    end:
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Jul 12 2014

T[ n_, k_] := QBinomial[n, k, -1]; (* Michael Somos, Jun 14 2011; since V7 *)
Clear[p, n, x, a]
w = 0;
p[x, 1] := 1;
p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + w*x + 1)^Floor[n/2]]
a = Table[CoefficientList[p[x, n], x], {n, 1, 12}]
Flatten[a] (* Roger L. Bagula and Gary W. Adamson, Dec 04 2009 *)

{T(n, k) = binomial(n%2, k%2) * binomial(n\2, k\2)};

from math import comb as binomial
def T(n, k): return binomial(n%2, k%2) * binomial(n//2, k//2)
print([T(n, k) for n in range(14) for k in range(n+1)])  # Peter Luschny, Oct 17 2024

@cached_function
def T(n, k):
    if k == 0 or k == n: return 1
    return T(n-1, k-1) + (-1)^k*T(n-1, k)
for n in (0..12): print([T(n, k) for k in (0..n)]) # Peter Luschny, Jul 06 2021

T(n, k) = T(n-1, k-1) + T(n-1, k) if n odd or k even, else 0. T(0, 0) = 1.
T(n, k) = T(n-2, k-2) + T(n-2, k). T(0, 0) = T(1, 0) = T(1, 1) = 1.
Square array made by setting first row/column to 1's (A(i, 0) = A(0, j) = 1); A(1, 1) = 0; A(1, j) = A(1, j-2); A(i, 1) = A(i-2, 1); other entries A(i, j) = A(i-2, j) + A(i, j-2). - Gerald McGarvey, Aug 21 2004
Sum_{k=0..n} k * T(n,k) = A093968(n); A093968 = S-D transform of A001477. - Philippe Deléham, Aug 02 2006
Equals 2*A034851 - A007318. - Gary W. Adamson, Dec 31 2007. [Corrected by Yosu Yurramendi, Aug 07 2008]
A051160(n, k) = (-1)^floor(k/2) * T(n, k).
Sum_{k = 0..n} T(n,k)*x^k = A000012(n), A016116(n+1), A056487(n), A136859(n+2) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Mar 11 2014
G.f.: (1+x+x*y)/(1-x^2-y^2*x^2). - Philippe Deléham, Mar 11 2014
For n,k >= 1, T(n, k) = 0 when n odd and k even; otherwise, T(n, k) = binomial(floor((n-1)/2), floor((k-1)/2)). - Christian Barrientos, Mar 14 2020
From Werner Schulte, Jun 25 2021: (Start)
T(n,k) = T(n-1,k-1) + (-1)^k * T(n-1,k) for 0 < k < n with initial values T(n,0) = T(n,n) = 1 for n >= 0.
Matrix inverse is T^-1(n,k) = (-1)^((n-k)*(n+k+1)/2) * T(n,k) for 0 <= k <= n. (End)
From Peter Bala, Aug 08 2021: (Start)
Double Riordan array ( 1/(1 - x); x/(1 + x), x/(1 - x) ) in the notation of Davenport et al.
G.f. for column 2*n: (1 + x)*x^(2*n)/(1 - x^2)^(n+1); G.f. for column 2*n+1: x^(2*n+1)/(1 - x^2)^(n+1)
Row polynomials: R(2*n,x) = (1 + x^2)^n; R(2*n+1,x) = (1 + x)*(1 + x^2)^n.
The infinitesimal generator of this triangle has the sequence [1, 0, 1, 0, 1, 0, ...] on the main subdiagonal, the sequence [1, 1, 2, 2, 3, 3, 4, 4, ...] on the diagonal immediately below and zeros elsewhere.
Let T denote this lower triangular array. Then T^a, for a in C, is the double Riordan array ( (1 + a*x)/(1 - a*x^2); x/(1 + a*x), (1 + a*x)/(1 - a*x^2) ) with o.g.f. (1 + x*(a + y))/(1 - x^2*(a + y^2)) =  1 + (a + y)*x + (a + y^2)*x^2 + (a^2 + a*y + a*y^2 + y^3)*x^3 + (a^2 + 2*a*y^2 + y^4)*x^4 + ....
The (2*n)-th row polynomial of T^a is (a + y^2)^n; The (2*n+1)-th row polynomial of T^a is (a + y)*(a + y^2)^n. (End)

New name using a formula of the author by Peter Luschny, Oct 17 2024

cp's OEIS Frontend

A096252 Array read by rows, starting with n=0: row n lists A057077(n+1)*8^(n+1)/2, A057077(n+2)*8^(n+1)/2, A057077(n+1)*8^(n+1).

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A000009 Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts.

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A007814 Exponent of highest power of 2 dividing n, a.k.a. the binary carry sequence, the ruler sequence, or the 2-adic valuation of n.

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A000312 a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

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A039599 Triangle formed from even-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

A096252 Array read by rows, starting with n=0: row n lists A057077(n+1)8^(n+1)/2, A057077(n+2)8^(n+1)/2, A057077(n+1)*8^(n+1).

A006950 G.f.: Product_{k>=1} (1 + x^(2k - 1)) / (1 - x^(2k)).