A051162 -id:A051162 - 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)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.
George E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.
George E. Andrews, Number Theory, Dover Publications, 1994, Theorem 12-3, pp. 154-5, and (13-1-1) p. 163.
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.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 99.
William Dunham, The Mathematical Universe, pp. 57-62, J. Wiley, 1994.
Leonhard Euler, De partitione numerorum, Novi commentarii academiae scientiarum Petropolitanae 3 (1750/1), 1753, reprinted in: Commentationes Arithmeticae. (Opera Omnia. Series Prima: Opera Mathematica, Volumen Secundum), 1915, Lipsiae et Berolini, 254-294.
Ian P. Goulden and David M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.1).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 86.
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.
Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 253.
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.
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
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.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002. [Local copy, with permission]
H. B. C. Darling, Collected Papers of Ramanujan, Table for q(n); n=1 through 100.
Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, Journal of Discrete Algorithms, Vol. 16 (2012), pp. 258-269; arXiv preprint, arXiv:1110.5103 [math.CO], 2011.
John A. Ewell, Partition recurrences, J. Comb. Theory A, Vol. 14, 125-127, 1973.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009; see pages 48 and 499.
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.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 108.
Martin Klazar, What is an answer? - remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Vaclav Kotesovec, Getting wrong limit with Bessel, Mathematica Stack Exchange, Nov 09 2016.
Alain Lascoux, Sylvester's bijection between strict and odd partitions, Discrete Math., Vol. 277, No. 1-3 (2004), pp. 275-278.
Jeremy Lovejoy, The number of partitions into distinct parts modulo powers of 5, Bulletin of the London Mathematical Society, Vol. 35, No. 1 (2003), pp. 41-46; alternative link.
James Mc Laughlin, Andrew V. Sills, and Peter Zimmer, Rogers-Ramanujan-Slater Type Identities, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008; see also arXiv version, arXiv:1901.00946 [math.NT], 2019.
Günter Meinardus, Über Partitionen mit Differenzenbedingungen, Mathematische Zeitschrift (1954/55), Volume 61, page 289-302.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q(n).
Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, Vol. 44, No. 2 (2017), pp. 417-435; alternative link.
István Mező, Several special values of Jacobi theta functions arXiv:1106.2703v3 [math.CA], 2011-2013.
Donald J. Newman, A Problem Seminar, pp. 18;93;102-3 Prob. 93 Springer-Verlag NY 1982.
Hieu D. Nguyen and Douglas Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence.
Kimeu Arphaxad Ngwava, Nick Gill, and Ian Short, Nilpotent covers of symmetric groups, arXiv:2005.13869 [math.GR], 2020.
Matthew Parker, The first 100K terms (7-Zip compressed file).
Marko Riedel, Proof of recurrence by V. Jovovic.
Ed Sandifer, How Euler Did It, Philip Naude's problem.
Michael Somos, Introduction to Ramanujan theta functions.
Zhi-Wei Sun, A representation problem involving strict partition numbers, Question 444761 at MathOverflow, April 14, 2023.
Eric Weisstein's World of Mathematics, Partition Function P, Partition Function Q, Partition Function bk, Euler Identity, Ramanujan Theta Functions, q-Pochhammer Symbol.
Wikipedia, Glaisher's Theorem.
Wolfram Research, Generating functions for q(n).
Mingjia Yang and Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.
Michael P. Zaletel and Roger S. K. Mong, Exact matrix product states for quantum Hall wave functions, Physical Review B, Vol. 86, No. 24 (2012), 245305; arXiv preprint, arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012
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

A000567 Octagonal numbers: n(3n-2). Also called star numbers.

Original entry on oeis.org

0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, 2640, 2821, 3008, 3201, 3400, 3605, 3816, 4033, 4256, 4485, 4720, 4961, 5208, 5461

Offset: 0

N. J. A. Sloane

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0,1,....
The spiral begins:
.
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
       90  60  36  18   6   0   3  12  27  48  75
       /   /   /   /   /   /   /   /   /   /   /
     91  61  37  19   7   1---2  11  26  47  74
       \   \   \   \   \ .       /   /   /   /
       92  62  38  20   8---9--10  25  46  73
         \   \   \   \ .           /   /   /
         93  63  39  21--22--23--24  45  72
           \   \   \ .               /   /
           94  64  40--41--42--43--44  71
             \   \ .                   /
             95  65--66--67--68--69--70
               \ .
               96
.
(End)
From Lekraj Beedassy, Oct 02 2003: (Start)
Also the number of distinct three-cell blocks that may be removed out of A000217(n+1) square cells arranged in a stepping triangular array of side (n+1). A 5-layer triangular array of square cells, for instance, has vertices outlined thus:
  x x
  x x x
  x x x x
  x x x x x
  x x x x x x
  x x x x x x  (End)
First derivative at n of A045991. - Ross La Haye, Oct 23 2004
Starting from n=1, the sequence corresponds to the Wiener index of K_{n,n} (the complete bipartite graph wherein each independent set has n vertices). - Kailasam Viswanathan Iyer, Mar 11 2009
Number of divisors of 24^(n-1) for n > 0 (cf A009968). - J. Lowell, Aug 30 2008
a(n) = A001399(6n-5), number of partitions of 6*n - 5 into parts < 4. For example a(2)=8 and partitions of 6*2 - 5 = 7 into parts < 4 are: [1,1,1,1,1,1,1], [1,1,1,1,1,2],[1,1,1,1,3], [1,1,1,2,2], [1,1,2,3], [1,2,2,2], [1,3,3], [2,2,3]. - Adi Dani, Jun 07 2011
Also, sequence found by reading the line from 0 in the direction 0, 8, ..., and the parallel line from 1 in the direction 1, 21, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Sep 10 2011
Partial sums give A002414. - Omar E. Pol, Jan 12 2013
Generate a Pythagorean triple using Euclid's formula with (n, n-1) to give A,B,C. a(n) = B + (A + C)/2. - J. M. Bergot, Jul 13 2013
The number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n-4; {1, 2n-2, 3, 2n-2, 1, 18n-8}]. For n=1, this collapses to [5; {5, 10}]. - Magus K. Chu, Oct 10 2022
a(n)*a(n+1) + 1 = (3n^2 + n - 1)^2. In general, a(n)*a(n+k) + k^2 = (3n^2 + (3k-2)n - k)^2. - Charlie Marion, May 23 2023

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 38.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1.
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 19-20.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

T. D. Noe, Table of n, a(n) for n = 0..1000
Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.
Cesar Ceballos and Viviane Pons, The s-weak order and s-permutahedra II: The combinatorial complex of pure intervals, arXiv:2309.14261 [math.CO], 2023. See p. 42.
C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.
John Elias, Illustration: Hexagonal spiral grids based on generalized octagonal numbers; Illustration: Generalized Square-Octagonal Grids.
John Elias, Illustration: Nesting Cube-frames; Nesting Cube Animation; Nesting-frames Decomposition; Factorial Compartmentalization.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 342.
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234. See Table 1.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Kaie Kubjas, Luca Sodomaco, and Elias Tsigaridas, Exact solutions in low-rank approximation with zeros, arXiv:2010.15636 [math.AG], 2020.
Viktor Levandovskyy, Christoph Koutschan, and Oleksandr Motsak, On Two-generated Non-commutative Algebras Subject to the Affine Relation, arXiv:1108.1108 [cs.SC], 2011.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567.
Leo Tavares, Illustration: Square Rays.
Leo Tavares, Illustration: Twin Rectangular Rays.
Leo Tavares, Illustration: Star Rows.
Leo Tavares, Illustration: Split Stars.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Octagonal Number.
Eric Weisstein's World of Mathematics, Wiener Index.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014641, A014642, A014793, A014794, A001835, A016777, A045944, A093563 ((6, 1) Pascal, column m=2). A016921 (differences).
Cf. A005408 (the odd numbers).
Cf. A000290, A000217, A046092, A000384, A002378, A003154.

List([0..50], n -> n*(3*n-2)); # G. C. Greubel, Nov 15 2018

a000567 n = n * (3 * n - 2)  -- Reinhard Zumkeller, Dec 20 2012

[n*(3*n-2) : n in [0..50]]; // Wesley Ivan Hurt, Oct 10 2021

A000567 := proc(n)
    n*(3*n-2) ;
end proc:
seq(A000567(n), n=1..50) ;

Table[n (3 n - 2), {n, 0, 50}] (* Harvey P. Dale, May 06 2012 *)
Table[PolygonalNumber[RegularPolygon[8], n], {n, 0, 43}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
PolygonalNumber[8, Range[0, 20]] (* Eric W. Weisstein, Sep 07 2017 *)
LinearRecurrence[{3, -3, 1}, {1, 8, 21}, {0, 20}] (* Eric W. Weisstein, Sep 07 2017 *)

a(n)=n*(3*n-2) \\ Charles R Greathouse IV, Jun 10 2011

vector(50, n, n--; n*(3*n-2)) \\ G. C. Greubel, Nov 15 2018

# Intended to compute the initial segment of the sequence, not isolated terms.
def aList():
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 6, y + 6
A000567 = aList()
print([next(A000567) for i in range(49)]) # Peter Luschny, Aug 04 2019

[n*(3*n-2) for n in range(50)] # Gennady Eremin, Mar 10 2022

[n*(3*n-2) for n in range(50)] # G. C. Greubel, Nov 15 2018

a(n) = n*(3*n-2).
a(n) = (3n-2)*(3n-1)*(3n)/((3n-1) + (3n-2) + (3n)), i.e., (the product of three consecutive numbers)/(their sum). a(1) = 1*2*3/(1+2+3), a(2) = 4*5*6/(4+5+6), etc. - Amarnath Murthy, Aug 29 2002
E.g.f.: exp(x)*(x+3*x^2). - Paul Barry, Jul 23 2003
G.f.: x*(1+5*x)/(1-x)^3. Simon Plouffe in his 1992 dissertation
a(n) = Sum_{k=1..n} (5*n - 4*k). - Paul Barry, Sep 06 2005
a(n) = n + 6*A000217(n-1). - Floor van Lamoen, Oct 14 2005
a(n) = C(n+1,2) + 5*C(n,2).
Starting (1, 8, 21, 40, 65, ...) = binomial transform of [1, 7, 6, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=8. - Jaume Oliver Lafont, Dec 02 2008
a(n) = A000578(n) - A007531(n). - Reinhard Zumkeller, Sep 18 2009
a(n) = a(n-1) + 6*n - 5 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010
a(n) = 2*a(n-1) - a(n-2) + 6. - Ant King, Sep 01 2011
a(n) = A000217(n) + 5*A000217(n-1). - Vincenzo Librandi, Nov 20 2010
a(n) = (A185212(n) - 1) / 4. - Reinhard Zumkeller, Dec 20 2012
a(n) = A174709(6n). - Philippe Deléham, Mar 26 2013
a(n) = (2*n-1)^2 - (n-1)^2. - Ivan N. Ianakiev, Apr 10 2013
a(6*a(n) + 16*n + 1) = a(6*a(n) + 16*n) + a(6*n + 1). - Vladimir Shevelev, Jan 24 2014
a(0) = 0, a(n) = Sum_{k=0..n-1} A005408(A051162(n-1,k)), n >= 1. - L. Edson Jeffery, Jul 28 2014
Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi + 9*log(3))/12 = 1.2774090575596367311949534921... . - Vaclav Kotesovec, Apr 27 2016
From Ilya Gutkovskiy, Jul 29 2016: (Start)
Inverse binomial transform of A084857.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(2*sqrt(3)) = A093766. (End)
a(n) = n * A016777(n-1) = A053755(n) - A000290(n+1). - Bruce J. Nicholson, Aug 10 2017
Product_{n>=2} (1 - 1/a(n)) = 3/4. - Amiram Eldar, Jan 21 2021
P(4k+4,n) = ((k+1)*n - k)^2 - (k*n - k)^2 where P(m,n) is the n-th m-gonal number (a generalization of the Apr 10 2013 formula, a(n) = (2*n-1)^2 - (n-1)^2). - Charlie Marion, Oct 07 2021
From Leo Tavares, Oct 31 2021: (Start)
a(n) = A000290(n) + 4*A000217(n-1). See Square Rays illustration.
a(n) = A000290(n) + A046092(n-1)
a(n) = A000384(n) + 2*A000217(n-1). See Twin Rectangular Rays illustration.
a(n) = A000384(n) + A002378(n-1)
a(n) = A003154(n) - A045944(n-1). See Star Rows illustration. (End)

Incorrect example removed by Joerg Arndt, Mar 11 2010

A025581 Triangle read by rows: T(n, k) = n-k, for 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3

Offset: 0

David W. Wilson

Decreasing integers m to 0 followed by decreasing integers m+1 to 0, etc.
The point with coordinates (x = A025581(n), y = A002262(n)) sweeps out the first quadrant by upwards antidiagonals. N. J. A. Sloane, Jul 17 2018
The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 0, k >= 0) by antidiagonals upwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002
Riordan array (x/(1-x)^2, x). - Philippe Deléham, Feb 18 2012
a(n,k) = (A214604(n,k) - A214661(n,k)) / 2. - Reinhard Zumkeller, Jul 25 2012
Sequence B is called a reverse reluctant sequence of sequence A if B is a triangular array read by rows such that row number k lists the first k terms of the sequence A in reverse order. This sequence is the reverse reluctant sequence of sequence 0,1,2,3,..., the nonnegative integers A001477. - Boris Putievskiy, Dec 13 2012
A problem posed by François Viète (Vieta) in his book Zeteticorum liber quinque (1593), liber 2, problem 19 (quoted in the Alten et al. reference, on p. 292) is to find for a rectangle (a >= b >= 1) with given a^3 - b^3, name it C, and a*b, name it F, the difference a-b, name it x. This is a simple exercise which Viète found remarkable. It reduces to a standard cubic equation for x, namely  x^3 + 3*F*x = C. Proof: Use the square of the diagonal d^2 = a^2 + b^2. Then (i) C = a^3 - b^3 = (a - b)*(a^2 + b^2 + a*b) = x*(d^2 + F). (ii) use the trivial relation d^2 = (a-b)^2 + 2*a*b = x^2 + 2*F, to eliminate d^2 in (i). End of the Proof. Here for positive integers a = n and b = k: (T(n, k)^2 + 3*A079904(n, k))*T(n, k) = A257238(n, k) (also true for n = k = 0). - Wolfdieter Lang, May 12 2015
See a comment on A051162 on the cubic equation for S = a+b in terms of Cplus = a^3 + b^3 and D = a - b. This equation leads to a - b = sqrt((4*Cplus -S^3)/(3*S)). - Wolfdieter Lang, May 15 2015
The entries correspond to the first of the 2 coordinates of the Cantor Pairs, specifically x=w-(CPKey-(w^2+w)/2), where w=floor((sqrt(8*CPKey+1)-1)/2) and CPKey=Cantor Pair key (A001477). The second of the coordinate pairs is A002262. - Bill McEachen, Sep 12 2015

			The triangle T(n, k) begins (note that one could use l <= k <= n, for any integer l, especially 1):
  n\k  0 1 2 3 4 5 6 7 8 9 10 ...
   0:  0
   1:  1 0
   2:  2 1 0
   3:  3 2 1 0
   4:  4 3 2 1 0
   5:  5 4 3 2 1 0
   6:  6 5 4 3 2 1 0
   7:  7 6 5 4 3 2 1 0
   8:  8 7 6 5 4 3 2 1 0
   9:  9 8 7 6 5 4 3 2 1 0
  10: 10 9 8 7 6 5 4 3 2 1 0
  ... [formatted by _Wolfdieter Lang_, May 12 2015]

H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, p. 203.

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Michael Somos, Sequences used for indexing triangular or square arrays
Eric Weisstein's World of Mathematics, Pairing Function.

Cf. A025669, A025676, A025683, A002262, A004736, A001477.
Cf. A141418 (partial sums per row).
Cf. A079904, A257238, A051162, A105125.

a025581 n k = n - k
a025581_row n = [n, n-1 .. 0]
a025581_tabl = iterate (\xs@(x:_) -> (x + 1) : xs) [0]
-- Reinhard Zumkeller, Aug 04 2014, Jul 22 2012, Mar 07 2011

/* As triangle */ [[(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Sep 13 2015

A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1): seq(A025581(n), n=0..100);

Flatten[NestList[Prepend[#, #[[1]]+1]&, {0}, 13]] (* Jean-François Alcover, May 17 2011 *)
With[{nn=20},Flatten[Table[Join[{0},Reverse[Range[i]]],{i,nn}]]] (* Harvey P. Dale, Dec 31 2014 *)
Table[Range[n,0,-1],{n,0,15}]//Flatten (* Harvey P. Dale, Aug 01 2020 *)

a(n)=binomial(1+floor(1/2+sqrt(2+2*n)),2)-(n+1) /* produces a(n) */

t1(n)=binomial(floor(3/2+sqrt(2+2*n)),2)-(n+1) /* A025581 */

t2(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2) /* A002262 */

apply( {A025581(n)=binomial(sqrtint(8*n+1)\/2+1,2)-n-1}, [0..90]) \\ M. F. Hasler, Dec 06 2019

from math import isqrt, comb
def A025581(n): return comb((m:=isqrt(k:=n+1<<1))+(k>m*(m+1))+1,2)-n-1 # Chai Wah Wu, Nov 08 2024

T(n, k) = n-k, for 0 <= k <= n.
As a sequence: a(n) = (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), with trinv(n) = floor((1+sqrt(1+8*n))/2). Cf. A002262.
a(n) = A004736(n+1) - 1.
G.f. for T(n,k): y / ((1-x)^2 * (1-x*y)). - Ralf Stephan, Jan 25 2005
For the cubic equation satisfied by T(n, k) see the comment on a problem by Viète above. - Wolfdieter Lang, May 12 2015
G.f. for a(n): -(1-x)^(-2) + (1-x)^(-1) * Sum_{n>=0} (n+1)*x^(n*(n+1)/2). The sum is related to Jacobi theta functions. - Robert Israel, May 12 2015
T(n, k) = sqrt((4*A105125(n, k) - A051162(n, k)^3)/(3*A051162(n, k))). See a comment above. - Wolfdieter Lang, May 15 2015
a(n) = (1/2)*(t^2 + t - 2*n - 2), where t = floor(sqrt(2*n+1) + 1/2) = round(sqrt(2*n+1)). - Ridouane Oudra, Dec 01 2019
a(n) = ((1/2) * ceiling((-1 + sqrt(9 + 8 * n))/2) * ceiling((1 + sqrt(9 + 8 * n))/2)) - n - 1. - Ryan Jean, Apr 22 2022

Typo in definition corrected by Arkadiusz Wesolowski, Nov 24 2011

Edited (part of name moved to first comment; definition of trinv added in formula) by Wolfdieter Lang, May 12 2015

A045943 Triangular matchstick numbers: a(n) = 3n(n+1)/2.

Original entry on oeis.org

0, 3, 9, 18, 30, 45, 63, 84, 108, 135, 165, 198, 234, 273, 315, 360, 408, 459, 513, 570, 630, 693, 759, 828, 900, 975, 1053, 1134, 1218, 1305, 1395, 1488, 1584, 1683, 1785, 1890, 1998, 2109, 2223, 2340, 2460, 2583, 2709, 2838, 2970, 3105, 3243, 3384, 3528

Offset: 0

R. K. Guy

Also, 3 times triangular numbers, a(n) = 3*A000217(n).
In the 24-bit RGB color cube, the number of color-lattice-points in r+g+b = n planes at n < 256 equals the triangular numbers. For n = 256, ..., 765 the number of legitimate color partitions is less than A000217(n) because {r,g,b} components cannot exceed 255. For n = 256, ..., 511, the number of non-color partitions are computable with A045943(n-255), while for n = 512, ..., 765, the number of color points in r+g+b planes equals A000217(765-n). - Labos Elemer, Jun 20 2005
If a 3-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-3) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
a(n) is also the smallest number that may be written both as the sum of n-1 consecutive positive integers and n consecutive positive integers. - Claudio Meller, Oct 08 2010
For n >= 3, a(n) equals 4^(2+n)*Pi^(1 - n) times the coefficient of zeta(3) in the following integral with upper bound Pi/4 and lower bound 0: int x^(n+1) tan x dx. - John M. Campbell, Jul 17 2011
The difference a(n)-a(n-1) = 3*n, for n >= 1. - Stephen Balaban, Jul 25 2011 [Comment clarified by N. J. A. Sloane, Aug 01 2024]
Sequence found by reading the line from 0, in the direction 0, 3, ..., and the same line from 0, in the direction 0, 9, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. This is one of the orthogonal axes of the spiral; the other is A032528. - Omar E. Pol, Sep 08 2011
A005449(a(n)) = A000332(3n + 3) = C(3n + 3, 4), a second pentagonal number of triangular matchstick number index number. Additionally, a(n) - 2n is a pentagonal number (A000326). - Raphie Frank, Dec 31 2012
Sum of the numbers from n to 2n. - Wesley Ivan Hurt, Nov 24 2015
Number of orbits of Aut(Z^7) as function of the infinity norm (n+1) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 5376 or 17920 or 20160. - Philippe A.J.G. Chevalier, Dec 28 2015
Also the number of 4-cycles in the (n+4)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
Number of terms less than 10^k, k=0,1,2,3,...: 1, 3, 8, 26, 82, 258, 816, 2582, 8165, 25820, 81650, 258199, 816497, 2581989, 8164966, ... - Muniru A Asiru, Jan 24 2018
Numbers of the form 3*m*(2*m + 1) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018
Partial sums of A008585. - Omar E. Pol, Jun 20 2018
Column 1 of A273464. (Number of ways to select a unit lozenge inside an isosceles triangle of side length n; all vertices on a hexagonal lattice.) - R. J. Mathar, Jul 10 2019
Total number of pips in the n-th suit of a double-n domino set. - Ivan N. Ianakiev, Aug 23 2020

			From _Stephen Balaban_, Jul 25 2011: (Start)
T(n), the triangular numbers = number of nodes,
a(n-1) = number of edges in the T(n) graph:
       o    (T(1) = 1, a(0) = 0)
       o
      / \   (T(2) = 3, a(1) = 3)
     o - o
       o
      / \
     o - o  (T(3) = 6, a(2) = 9)
    / \ / \
   o - o - o
... [Corrected by _N. J. A. Sloane_, Aug 01 2024] (End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 543.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 29.
T. Aaron Gulliver, Sums of Powers of Integers Divisible by Three, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, 1895 - 1901.
Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067
Milan Janjic, Two Enumerative Functions
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
E. Lábos, On the number of RGB-colors we can distinguish. Partition Spectra. Lecture at 7th Hungarian Conference on Biometry and Biomathematics. Budapest. Jul 06, 2005. Applied Ecology and Environmental Research 4(2): 159-169, 2006.
R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005448, A002378, A046092, A051162, A126804, A001318, A032528.
3 times n-gonal numbers: A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
A diagonal of A010027.
Orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A115067, A008585, A005843, A001477, A000217.
Cf. A027480 (partial sums).
Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
This sequence: Sum_{k = n..2*n} k.
Cf. A304993:   Sum_{k = n..2*n} k*(k+1)/2.
Cf. A050409:   Sum_{k = n..2*n} k^2.
Similar sequences are listed in A316466.

List([0..10^4], n -> 3*n*(n+1)/2); # Muniru A Asiru, Jan 24 2018

a n = sum [x | x <- [n..2*n]] -- Peter Kagey, Jul 27 2015

[3*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, May 02 2011

seq(3*binomial(n+1,2), n=0..49); # Zerinvary Lajos, Nov 24 2006

Table[3 n (n + 1)/2, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Oct 31 2008 *)
3 Accumulate@Range[0, 48] (* Arkadiusz Wesolowski, Oct 29 2012 *)
CoefficientList[Series[-3 x/(x - 1)^3, {x, 0, 47}], x] (* Robert G. Wilson v, Jan 29 2015 *)
LinearRecurrence[{3, -3, 1}, {0, 3, 9}, 50] (* Jean-François Alcover, Dec 12 2016 *)

a(n)=3*binomial(n+1,2) \\ Charles R Greathouse IV, Jun 16 2011

(3 to 150 by 3).scanLeft(0)( + ) // Alonso del Arte, Sep 12 2019

a(n) is the sum of n+1 integers starting from n, i.e., 1+2, 2+3+4, 3+4+5+6, 4+5+6+7+8, etc. - Jon Perry, Jan 15 2004
a(n) = A126890(n+1,n-1) for n>1. - Reinhard Zumkeller, Dec 30 2006
a(n) + A145919(3*n+3) = 0. - Matthew Vandermast, Oct 28 2008
a(n) = A000217(2*n) - A000217(n-1); A179213(n) <= a(n). - Reinhard Zumkeller, Jul 05 2010
a(n) = a(n-1)+3*n, n>0. - Vincenzo Librandi, Nov 18 2010
G.f.: 3*x/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = A005448(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(n) = A001477(n)+A000290(n)+A000217(n). - J. M. Bergot, Dec 08 2012
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2. - Wesley Ivan Hurt, Nov 24 2015
a(n) = A027480(n)-A027480(n-1). - Peter M. Chema, Jan 18 2017.
2*a(n)+1 = A003215(n). - Miquel Cerda, Jan 22 2018
a(n) = T(2*n) - T(n-1), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - Charlie Marion, Dec 06 2020
E.g.f.: 3*exp(x)*x*(2 + x)/2. - Stefano Spezia, May 19 2021
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 2/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(2*log(2)-1)/3. (End)
Product_{n>=1} (1 - 1/a(n)) = -(3/(2*Pi))*cos(sqrt(11/3)*Pi/2). - Amiram Eldar, Feb 21 2023

A085478 Triangle read by rows: T(n, k) = binomial(n + k, 2*k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 15, 7, 1, 1, 15, 35, 28, 9, 1, 1, 21, 70, 84, 45, 11, 1, 1, 28, 126, 210, 165, 66, 13, 1, 1, 36, 210, 462, 495, 286, 91, 15, 1, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 1, 55, 495, 1716, 3003, 3003, 1820, 680, 153, 19, 1

Offset: 0

Philippe Deléham, Aug 14 2003

Coefficient array for Morgan-Voyce polynomial b(n,x). A053122 (unsigned) is the coefficient array for B(n,x). Reversal of A054142. - Paul Barry, Jan 19 2004
This triangle is formed from even-numbered rows of triangle A011973 read in reverse order. - Philippe Deléham, Feb 16 2004
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k+1 peaks. T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k peaks at height >= 2. T(n,k) is the number of directed column-convex polyominoes of area n+1, having k+1 columns. - Emeric Deutsch, May 31 2004
Riordan array (1/(1-x), x/(1-x)^2). - Paul Barry, May 09 2005
The triangular matrix a(n,k) = (-1)^(n+k)*T(n,k) is the matrix inverse of A039599. - Philippe Deléham, May 26 2005
The n-th row gives absolute values of coefficients of reciprocal of g.f. of bottom-line of n-wave sequence. - Floor van Lamoen (fvlamoen(AT)planet.nl), Sep 24 2006
Unsigned version of A129818. - Philippe Deléham, Oct 25 2007
T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k >=1 (height(alpha) = |Im(alpha)|) and of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Oct 02 2008
A085478 is jointly generated with A078812 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n>1, u(n,x) = u(n-1,x)+x*v(n-1)x and v(n,x) = u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 25 2012
Per Kimberling's recursion relations, see A102426. - Tom Copeland, Jan 19 2016
Subtriangle of the triangle given by (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012
T(n,k) is also the number of compositions (ordered partitions) of 2*n+1 into 2*k+1 parts which are all odd. Proof: The o.g.f. of column k, x^k/(1-x)^(2*k+1) for k >= 0, is the o.g.f. of the odd-indexed members of the sequence with o.g.f. (x/(1-x^2))^(2*k+1) (bisection, odd part). Thus T(n,k) is obtained from the sum of the multinomial numbers A048996 for the partitions of 2*n+1 into 2*k+1 parts, all of which are odd. E.g., T(3,1) = 3 + 3 from the numbers for the partitions [1,1,5] and [1,3,3], namely 3!/(2!*1!) and 3!/(1!*2!), respectively. The number triangle with the number of these partitions as entries is A152157. - Wolfdieter Lang, Jul 09 2012
The matrix elements of the inverse are T^(-1)(n,k) = (-1)^(n+k)*A039599(n,k). - R. J. Mathar, Mar 12 2013
T(n,k) = A258993(n+1,k) for k = 0..n-1. - Reinhard Zumkeller, Jun 22 2015
The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the algebraic function F(x)*G(x)^n about 0, where F(x) = (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) and G(x) = ((1 + sqrt(1 + 4*x))/2)^2. For example, for n = 4, (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) * ((1 + sqrt(1 + 4*x))/2)^8 = (x^4  + 10*x^3 + 15*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 23 2018
Row n also gives the coefficients of the characteristc polynomial of the tridiagonal n X n matrix M_n given in A332602: Phi(n, x) := Det(M_n - x*1_n) = Sum_{k=0..n} T(n, k)*(-x)^k, for n >= 0, with Phi(0, x) := 1. - Wolfdieter Lang, Mar 25 2020
It appears that the largest root of the n-th degree polynomial is equal to the sum of the distinct diagonals of a (2*n+1)-gon including the edge, 1. The largest root of x^3 - 6*x^2 + 5*x - 1 is 5.048917... = the sum of (1 + 1.80193... + 2.24697...). Alternatively, the largest root of the n-th degree polynomial is equal to the square of sigma(2*n+1). Check: 5.048917... is the square of sigma(7), 2.24697.... Given N = 2*n+1, sigma(N) (N odd) can be defined as 1/(2*sin(Pi/(2*N))). Relating to the 9-gon, the largest root of x^4 - 10*x^3 + 15*x^2 - 7*x + 1 is 8.290859..., = the sum of (1 + 1.879385... + 2.532088... + 2.879385...), and is the square of sigma(9), 2.879385... Refer to A231187 for a further clarification of sigma(7). - Gary W. Adamson, Jun 28 2022
For n >=1, the n-th row is given by the coefficients of the minimal polynomial of -4*sin(Pi/(4*n + 2))^2. - Eric W. Weisstein, Jul 12 2023
Denoting this lower triangular array by L, then L * diag(binomial(2*k,k)^2) * transpose(L) is the LDU factorization of A143007, the square array of crystal ball sequences for the A_n X A_n lattices. - Peter Bala, Feb 06 2024
T(n, k) is the number of occurrences of the periodic substring (01)^k in the periodic string (01)^n (see Proposition 4.7 at page 7 in Fang). - Stefano Spezia, Jun 09 2024

			Triangle begins as:
  1;
  1    1;
  1    3    1;
  1    6    5    1;
  1   10   15    7    1;
  1   15   35   28    9    1;
  1   21   70   84   45   11    1;
  1   28  126  210  165   66   13    1;
  1   36  210  462  495  286   91   15    1;
  1   45  330  924 1287 1001  455  120   17    1;
  1   55  495 1716 3003 3003 1820  680  153   19    1;
...
From _Philippe Deléham_, Mar 26 2012: (Start)
(0, 1, 0, 1, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, ...) begins:
  1
  0, 1
  0, 1,  1
  0, 1,  3,   1
  0, 1,  6,   5,   1
  0, 1, 10,  15,   7,   1
  0, 1, 15,  35,  28,   9,  1
  0, 1, 21,  70,  84,  45, 11,  1
  0, 1, 28, 126, 210, 165, 66, 13, 1. (End)

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - _N. J. A. Sloane_, May 10 2012
Peter Bala, A 4-parameter family of embedded Riordan arrays
E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.
Paul Barry and Aoife Hennessy, Notes on a Family of Riordan Arrays and Associated Integer Hankel Transforms , JIS 12 (2009) 09.5.3.
Eduardo H. M. Brietzke, Recurrence Relations for Sums of Binomial Coefficients and Some Generalizations, Journal of Integer Sequences, Vol. 27 (2024), Article 24.3.4.
E. Czabarka et al, Enumerations of peaks and valleys on non-decreasing Dyck paths, Disc. Math. 341 (2018) 2789-2807, Theorem 3.
Emeric Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Wenjie Fang, Maximal number of subword occurrences in a word, arXiv:2406.02971 [math.CO], 2024.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62. - _Abdullahi Umar_, Oct 02 2008
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174.
Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, (2005) 359-370.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Row sums: A001519. Signed versions: A123970, A129818.
Cf. A000108, A001006, A007318, A098158, A183160, A078812, A091042.
Cf. A258993, A025581, A051162, A054142, A332602.
Cf. A231187, A143007.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n+k, 2*k) ))); # G. C. Greubel, Aug 01 2019

a085478 n k = a085478_tabl !! n !! k
a085478_row n = a085478_tabl !! n
a085478_tabl = zipWith (zipWith a007318) a051162_tabl a025581_tabl
-- Reinhard Zumkeller, Jun 22 2015

[Binomial(n+k, 2*k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

T := (n,k) -> binomial(n+k,2*k): seq(seq(T(n,k), k=0..n), n=0..11);

(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A085478 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A078812 *) (*Clark Kimberling, Feb 25 2012 *)
(* Second program *)
Table[Binomial[n + k, 2 k], {n, 0, 12}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 01 2019 *)
CoefficientList[Table[Fibonacci[2 n + 1, Sqrt[x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Jul 03 2023 *)
Join[{{1}}, CoefficientList[Table[MinimalPolynomial[-4 Sin[Pi/(4 n + 2)]^2, x], {n, 20}], x]] (* Eric W. Weisstein, Jul 12 2023 *)

PARI
```
T(n,k) = binomial(n+k,n-k)
```

[[binomial(n+k,2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

T(n, k) = (n+k)!/((n-k)!*(2*k)!).
G.f.: (1-z)/((1-z)^2-tz). - Emeric Deutsch, May 31 2004
Row sums are A001519 (Fibonacci(2n+1)). Diagonal sums are A011782. Binomial transform of A026729 (product of lower triangular matrices). - Paul Barry, Jun 21 2004
T(n, 0) = 1, T(n, k) = 0 if n=0} T(n-1-j, k-1)*(j+1). T(0, 0) = 1, T(0, k) = 0 if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j). For the column k, g.f.: Sum_{n>=0} T(n, k)*x^n = (x^k) / (1-x)^(2*k+1). - Philippe Deléham, Feb 15 2004
Sum_{k=0..n} T(n,k)*x^(2*k) = A000012(n), A001519(n+1), A001653(n), A078922(n+1), A007805(n), A097835(n), A097315(n), A097838(n), A078988(n), A097841(n), A097727(n), A097843(n), A097730(n), A098244(n), A097733(n), A098247(n), A097736(n), A098250(n), A097739(n), A098253(n), A097742(n), A098256(n), A097767(n), A098259(n), A097770(n), A098262(n), A097773(n), A098292(n), A097776(n) for x=0,1,2,...,27,28 respectively. - Philippe Deléham, Dec 31 2007
T(2*n,n) = A005809(n). - Philippe Deléham, Sep 17 2009
A183160(n) = Sum_{k=0..n} T(n,k)*T(n,n-k). - Paul D. Hanna, Dec 27 2010
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Feb 06 2012
O.g.f. for column k: x^k/(1-x)^(2*k+1), k >= 0. [See the o.g.f. of the triangle above, and a comment on compositions. - Wolfdieter Lang, Jul 09 2012]
E.g.f.: (2/sqrt(x + 4))*sinh((1/2)*t*sqrt(x + 4))*cosh((1/2)*t*sqrt(x)) = t + (1 + x)*t^3/3! + (1 + 3*x + x^2)*t^5/5! + (1 + 6*x + 5*x^2 + x^3)*t^7/7! + .... Cf. A091042. - Peter Bala, Jul 29 2013
T(n, k) = A065941(n+3*k, 4*k) = A108299(n+3*k, 4*k) = A194005(n+3*k, 4*k). - Johannes W. Meijer, Sep 05 2013
Sum_{k=0..n} (-1)^k*T(n,k)*A000108(k) = A000007(n) for n >= 0. - Werner Schulte, Jul 12 2017
Sum_{k=0..floor(n/2)} T(n-k,k)*A000108(k) = A001006(n) for n >= 0. - Werner Schulte, Jul 12 2017
From  Peter Bala, Jun 26 2025: (Start)
The n-th row polynomial b(n, x) = (-1)^n * U(2*n, (i/2)*sqrt(x)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.
b(n, x) = (-1)^n * Dir(n, -1 - x/2), where Dir(n, x) is the n-th row polynomial of the triangle A244419.
b(n, -1 - x) is the n-th row polynomial of A098493. (End)

A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

Original entry on oeis.org

1, 1, 3, 5, 15, 33, 91, 221, 583, 1465, 3795, 9653, 24831, 63441, 162763, 416525, 1067575, 2733673, 7003971, 17938661, 45954543, 117709185, 301527355, 772364093, 1978473511

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

Suppose that p=p(0)*x^n+p(1)*x^(n-1)+...+p(n-1)*x+p(n) is a polynomial of positive degree and that Q is a sequence of polynomials: q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k), for k=0,1,2,... The Q-downstep of p is the polynomial given by D(p)=p(0)*q(n-1,x)+p(1)*q(n-2,x)+...+p(n-1)*q(0,x)+p(n).

Since degree(D(p))

Example: let p(x)=2*x^3+3*x^2+4*x+5 and q(k,x)=(x+1)^k.

D(p)=2(x+1)^2+3(x+1)+4(1)+5=2x^2+7x+14

D(D(p))=2(x+1)+7(1)+14=2x+23

D(D(D(p)))=2(1)+23=25;

the Q-residue of p is 25.

We may regard the sequence Q of polynomials as the triangular array formed by coefficients:

t(0,0)

t(1,0)....t(1,1)

t(2,0)....t(2,1)....t(2,2)

t(3,0)....t(3,1)....t(3,2)....t(3,3)

and regard p as the vector (p(0),p(1),...,p(n)). If P is a sequence of polynomials [or triangular array having (row n)=(p(0),p(1),...,p(n))], then the Q-residues of the polynomials form a numerical sequence.

Following are examples in which Q is the triangle given by t(i,j)=1 for 0<=i<=j:

Q.....P...................Q-residue of P

1.....1...................A000079, 2^n

1....(x+1)^n..............A007051, (1+3^n)/2

1....(x+2)^n..............A034478, (1+5^n)/2

1....(x+3)^n..............A034494, (1+7^n)/2

1....(2x+1)^n.............A007582

1....(3x+1)^n.............A081186

1....(2x+3)^n.............A081342

1....(3x+2)^n.............A081336

1.....A040310.............A193649

1....(x+1)^n+(x-1)^n)/2...A122983

1....(x+2)(x+1)^(n-1).....A057198

1....(1,2,3,4,...,n)......A002064

1....(1,1,2,3,4,...,n)....A048495

1....(n,n+1,...,2n).......A087323

1....(n+1,n+2,...,2n+1)...A099035

1....p(n,k)=(2^(n-k))*3^k.A085350

1....p(n,k)=(3^(n-k))*2^k.A090040

1....A008288 (Delannoy)...A193653

1....A054142..............A101265

1....cyclotomic...........A193650

1....(x+1)(x+2)...(x+n)...A193651

1....A114525..............A193662

More examples:

Q...........P.............Q-residue of P

(x+1)^n...(x+1)^n.........A000110, Bell numbers

(x+1)^n...(x+2)^n.........A126390

(x+2)^n...(x+1)^n.........A028361

(x+2)^n...(x+2)^n.........A126443

(x+1)^n.....1.............A005001

(x+2)^n.....1.............A193660

A094727.....1.............A193657

(k+1).....(k+1)...........A001906 (even-ind. Fib. nos.)

(k+1).....(x+1)^n.........A112091

(x+1)^n...(k+1)...........A029761

(k+1)......A049310........A193663

(In these last four, (k+1) represents the triangle t(n,k)=k+1, 0<=k<=n.)

A051162...(x+1)^n.........A193658

A094727...(x+1)^n.........A193659

A049310...(x+1)^n.........A193664

A075362....A075362........A193665

Changing the notation slightly leads to the Mathematica program below and the following formulation for the Q-downstep of p: first, write t(n,k) as q(n,k). Define r(k)=Sum{q(k-1,i)*r(k-1-i) : i=0,1,...,k-1} Then row n of D(p) is given by v(n)=Sum{p(n,k)*r(n-k) : k=0,1,...,n}.

			First five rows of Q, coefficients of Fibonacci polynomials (A049310):
1
1...0
1...0...1
1...0...2...0
1...0...3...0...1
To obtain a(4)=15, downstep four times:
D(x^4+3*x^2+1)=(x^3+x^2+x+1)+3(x+1)+1: (1,1,4,5) [coefficients]
DD(x^4+3*x^2+1)=D(1,1,4,5)=(1,2,11)
DDD(x^4+3*x^2+1)=D(1,2,11)=(1,14)
DDDD(x^4+3*x^2+1)=D(1,14)=15.

Cf. A192872 (polynomial reduction), A193091 (polynomial augmentation), A193722 (the upstep operation and fusion of polynomial sequences or triangular arrays).

q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
f[n_, x_] := Fibonacci[n + 1, x];
p[n_, k_] := Coefficient[f[n, x], x, k]; (* A049310 *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 24}]    (* A193649 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]

Conjecture: G.f.: -(1+x)*(2*x-1) / ( (x-1)*(4*x^2+x-1) ). - R. J. Mathar, Feb 19 2015

A070770 b + c + d where b >= c >= d >= 0 ordered by b then c then d.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 4, 5, 6, 3, 4, 5, 5, 6, 7, 6, 7, 8, 9, 4, 5, 6, 6, 7, 8, 7, 8, 9, 10, 8, 9, 10, 11, 12, 5, 6, 7, 7, 8, 9, 8, 9, 10, 11, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 15, 6, 7, 8, 8, 9, 10, 9, 10, 11, 12, 10, 11, 12, 13, 14, 11, 12, 13, 14, 15, 16, 12, 13, 14, 15, 16, 17, 18, 7

Offset: 0

Henry Bottomley, May 06 2002

			Triangle begins:
  0,
  ;
  1;
  2, 3;
  ;
  2;
  3, 4;
  4, 5, 6;
  ;
  3;
  4, 5,
  5, 6, 7;
  6, 7, 8, 9;
  ;
  4;
  5, 6;
  6, 7,  8;
  7, 8,  9, 10;
  8, 9, 10, 11, 12;
  ;
  ...

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001477, A051162, A070771, A070772 for similar sequences with different numbers of terms summed.

seq(seq(seq(b+c+d,d=0..c),c=0..b),b=0..10); # Robert Israel, Jun 21 2018

for(x=0,5,for(y=0,x,for(z=0,y,print1(x+y+z", ")))) \\ Charles R Greathouse IV, Sep 17 2015

from math import isqrt, comb
from sympy import integer_nthroot
def A070770(n): return (m:=integer_nthroot(6*(n+1),3)[0])+(a:=n>=comb(m+2,3))+(k:=isqrt(b:=(c:=n+1-comb(m+a+1,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1)+c-2-comb(k+(b>k*(k+1)),2) # Chai Wah Wu, Dec 11 2024

a(n) = A056556(n) + A056557(n) + A056558(n).

A108872 Sums of ordinal references for a triangular table read by columns, top to bottom.

Original entry on oeis.org

2, 3, 4, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 16, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22

Offset: 1

Andrew S. Plewe, Jul 13 2005

The ordinal references (i,j) for a triangular table are arranged as follows:
  (1,1) (2,1) (3,1)
  ..... (2,2) (3,2)
  ........... (3,3)
The sequence comprises the sum of each reference in each column, read top to bottom. A similar sequence is A003057, which consists of the sums of the ordinal references for an array read by antidiagonals.
Subtriangle of triangle in A051162. - Philippe Deléham, Mar 26 2013
First 9 rows coincide with triangle A248110; T(n,k) = A002260(n,k) + n; T(2*n-1,n) = A016789(n-1). - Reinhard Zumkeller, Oct 01 2014

			a(1) = (1,1) = 1 + 1 = 2
a(2) = (2,1) = 2 + 1 = 3
a(3) = (2,2) = 2 + 2 = 4
a(4) = (3,1) = 3 + 1 = 4, etc.
Triangle begins:
  2
  3, 4
  4, 5, 6
  5, 6, 7, 8
  6, 7, 8, 9, 10
  7, 8, 9, 10, 11, 12
  8, 9, 10, 11, 12, 13, 14
  9, 10, 11, 12, 13, 14, 15, 16
  ... - _Philippe Deléham_, Mar 26 2013

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
NRICH discussion thread, Floor Function Identity. [Via Wayback Machine]

Cf. A003057.
Cf. A002024, A002260.
Cf. A005408, A005843.
Cf. A016789 (central terms), A248110.

a108872 n k = a108872_tabl !! (n-1) !! (k-1)
a108872_row n = a108872_tabl !! (n-1)
a108872_tabl = map (\x -> [x + 1 .. 2 * x]) [1..]
-- Reinhard Zumkeller, Oct 01 2014

Flatten[ Table[i + j, {j, 1, 12}, {i, 1, j}]] (* Jean-François Alcover, Oct 07 2011 *)

from math import isqrt
def A108872(n): return n+((r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-r)>>1) # Chai Wah Wu, Nov 08 2024

a(n) = a(i, j) = i + j
a(n) = A002024(n) + A002260(n) = floor(1/2 + sqrt(2n)) + n - (m(m+1)/2) + 1, where m = floor((sqrt(8n+1) - 1) / 2 ). The floor function may be computed directly by using the expression floor(x) = x + (arctan(cot(Pi*x)) / Pi) - 1/2 (equation from nrich.maths.org, see links).
Sum_{k=0..n} T(n,k) = A005449(n+1). - Philippe Deléham, Mar 26 2013

Offset changed by Reinhard Zumkeller, Oct 01 2014

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A100936 Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)1, C(n,1)2,..., C(n,k)A051163(k), ..., C(n,n)A051163(n)] and where A051162 equals the antidiagonal sums.

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