A003724 -id:A003724 - 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).
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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

A136630 Triangular array: T(n,k) counts the partitions of the set [n] into k odd sized blocks.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 1, 0, 10, 0, 1, 0, 0, 16, 0, 20, 0, 1, 0, 1, 0, 91, 0, 35, 0, 1, 0, 0, 64, 0, 336, 0, 56, 0, 1, 0, 1, 0, 820, 0, 966, 0, 84, 0, 1, 0, 0, 256, 0, 5440, 0, 2352, 0, 120, 0, 1, 0, 1, 0, 7381, 0, 24970, 0, 5082, 0, 165, 0, 1, 0, 0, 1024, 0, 87296, 0

Offset: 0

Paul D. Hanna, Jan 14 2008

For partitions into blocks of even size see A156289.
Essentially the unsigned matrix inverse of triangle A121408.
From Peter Bala, Jul 28 2014: (Start)
Define a polynomial sequence x_(n) by setting x_(0) = 1 and for n = 1,2,... setting x_(n) = x*(x + n - 2)*(x + n - 4)*...*(x + n - 2*(n - 1)). Then this table is the triangle of connection constants for expressing the monomial polynomials x^n in terms of the basis x_(k), that is, x^n = sum {k = 0..n} T(n,k)*x_(k) for n = 0,1,2,.... An example is given below.
Let M denote the lower unit triangular array A119467 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle, omitting the first row and column, equals the infinite matrix product M(0)*M(1)*M(2)*.... (End)
Also the Bell transform of A000035(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle begins:
  1;
  0, 1;
  0, 0,   1;
  0, 1,   0,    1;
  0, 0,   4,    0,    1;
  0, 1,   0,   10,    0,     1;
  0, 0,  16,    0,   20,     0,    1;
  0, 1,   0,   91,    0,    35,    0,    1;
  0, 0,  64,    0,  336,     0,   56,    0,   1;
  0, 1,   0,  820,    0,   966,    0,   84,   0,   1;
  0, 0, 256,    0, 5440,     0, 2352,    0, 120,   0, 1;
  0, 1,   0, 7381,    0, 24970,    0, 5082,   0, 165, 0, 1;
T(5,3) = 10. The ten partitions of the set [5] into 3 odd-sized blocks are
(1)(2)(345), (1)(3)(245), (1)(4)(235), (1)(5)(234), (2)(3)(145),
(2)(4)(135), (2)(5)(134), (3)(4)(125), (3)(5)(124), (4)(5)(123).
Connection constants: Row 5 = [0,1,0,10,0,1]. Hence, with the polynomial sequence x_(n) as defined in the Comments section we have x^5 = x_(1) + 10*x_(3) + x_(5) = x + 10*x*(x+1)*(x-1) + x*(x+3)*(x+1)*(x-1)*(x-3).

L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 225-226.

Ch. A. Charalambides, Central factorial numbers and related expansions, Fib. Quarterly, Vol. 19, No 5, Dec 1981, pp. 451-456.
Feng Qi and Peter Taylor, Series expansions for powers of sinc function and closed-form expressions for specific partial Bell polynomials, Appl. Anal. Disc. Math. (2024) Vol. 18, No. 1, 1-24. See p. 13.

Cf. A121408; A136631 (antidiagonal sums), A003724 (row sums), A136632; A002452 (column 3), A002453 (column 5); A008958 (central factorial triangle), A156289. A185690, A196776.

A136630 := proc (n, k) option remember; if k < 0 or n < k then 0 elif k = n then 1 else procname(n-2, k-2) + k^2*procname(n-2, k) end if end proc: seq(seq(A136630(n, k), k = 1 .. n), n = 1 .. 12); # Peter Bala, Jul 27 2014
# The function BellMatrix is defined in A264428.
BellMatrix(n -> (n+1) mod 2, 9); # Peter Luschny, Jan 27 2016

t[n_, k_] := Coefficient[ x^k/Product[ 1 - (2*j + k - 2*Quotient[k, 2])^2*x^2, {j, 0, k/2}] + x*O[x]^n, x, n]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2013, after Pari *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 13;
M = BellMatrix[Mod[#+1, 2]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

{T(n,k)=polcoeff(x^k/prod(j=0,k\2,1-(2*j+k-2*(k\2))^2*x^2 +x*O(x^n)),n)}

G.f. for column k: x^k/Product_{j=0..floor(k/2)} (1 - (2*j + k-2*floor(k/2))^2 * x^2).
G.f. for column 2*k: x^(2*k)/Product_{j=0..k} (1 - (2*j)^2*x^2).
G.f. for column 2*k+1: x^(2*k+1)/Product_{j=0..k} (1 - (2*j+1)^2*x^2).
From Peter Bala, Feb 21 2011 (Start)
T(n,k) = 1/(2^k*k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(2*j-k)^n,
Recurrence relation T(n+2,k) = T(n,k-2) + k^2*T(n,k).
E.g.f.: F(x,z) = exp(x*sinh(z)) = Sum_{n>=0} R(n,x)*z^n/n! = 1 + x*z + x^2*z^2/2! + (x+x^3)*z^3/3! + ....
The row polynomials R(n,x) begin
R(1,x) = x
R(2,x) = x^2
R(3,x) = x+x^3.
The e.g.f. F(x,z) satisfies the partial differential equation d^2/dz^2(F) = x^2*F + x*F' + x^2*F'' where ' denotes differentiation w.r.t. x.
Hence the row polynomials satisfy the recurrence relation R(n+2,x) = x^2*R(n,x) + x*R'(n,x) + x^2*R''(n,x) with R(0,x) = 1.
The recurrence relation for T(n,k) given above follows from this.
(End)
For the corresponding triangle of ordered partitions into odd-sized blocks see A196776. Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of exp(t*M) lists the row polynomials for the present triangle. - Peter Bala, Oct 06 2011
Row generating polynomials equal D^n(exp(x*t)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A196776. - Peter Bala, Dec 06 2011
From Peter Bala, Jul 28 2014: (Start)
E.g.f.: exp(t*sinh(x)) = 1 + t*x + t^2*x^2/2! + (t + t^3)*x^3/3! + ....
Hockey-stick recurrence: T(n+1,k+1) = Sum_{i = 0..floor((n-k)/2)} binomial(n,2*i)*T(n-2*i,k).
Recurrence equation for the row polynomials R(n,t):
R(n+1,t) = t*Sum_{k = 0..floor(n/2)} binomial(n,2*k)*R(n-2*k,t) with R(0,t) = 1. (End)

A006154 Number of labeled ordered partitions of an n-set into odd parts.

Original entry on oeis.org

1, 1, 2, 7, 32, 181, 1232, 9787, 88832, 907081, 10291712, 128445967, 1748805632, 25794366781, 409725396992, 6973071372547, 126585529106432, 2441591202059281, 49863806091395072, 1074927056650469527, 24392086908129247232, 581176736647853024581

Offset: 0

Simon Plouffe

Conjecture: taking the sequence modulo an integer k gives an eventually periodic sequence. For example, the sequence taken modulo 10 is [1, 1, 2, 7, 2, 1, 2, 7, 2, 1, 2, 7, 2, ...], with an apparent period [1, 2, 7, 2] beginning at a(1), of length 4. Cf. A000670. - Peter Bala, Apr 12 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Farid Aliniaeifard and Shu Xiao Li, Peak algebra in noncommuting variables, arXiv:2506.12868 [math.CO], 2025. See pp. 2, 9, 45.
Philippe Flajolet, Symbolic Enumerative Combinatorics and Complex Asymptotic Analysis, Algorithms Seminar 2000-2001, F. Chyzak (ed.), INRIA, (2002), pp. 161-170.
S. Getu and L. W. Shapiro, Combinatorial view of the composition of functions, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
Prabha Sivaraman Nair and Rejikumar Karunakaran, On k-Fibonacci Brousseau Sums, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 8.

Cf. A000045, A000111, A000670, A003724, A193474, A196776.

readlib(coeftayl):
with(combinat, bell);
A:=series(1/(1-sinh(x)),x,20);
G(x):=A : f[0]:=G(x): for n from 0 to 21 do f[n]:=coeftayl(G(x), x=0, n);;
p[n]:=f[n]*((n)!) od: x:=0:seq(p[n], n=0..20); # Sergei N. Gladkovskii, Jun 01 2012
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((i->
      a(n-i)*binomial(n, i))(2*j+1), j=0..(n-1)/2))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 01 2022

a[n_] := Sum[ (-1)^i*(k - 2*i)^n*Binomial[k, i]/2^k, {k, 1, n}, {i, 0, k}]; a[0] = 1; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Dec 07 2011, after Vladimir Kruchinin *)
With[{nn=20},CoefficientList[Series[1/(1-Sinh[x]),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Nov 16 2012 *)

a(n):=sum(sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k)/2^k,k,1,n); /* Vladimir Kruchinin, May 28 2011 */

a(n)=if(n<2,n>=0,sum(k=1,ceil(n/2),binomial(n,2*k-1)*a(n-2*k+1))) \\ Ralf Stephan

E.g.f.: 1/(1 - sinh(x)).
With alternating signs, e.g.f.: 1/(1+sinh(x)). - Ralf Stephan, Apr 29 2004
a(0) = a(1) = 1, a(n) = Sum_{k=1..ceiling(n/2)} C(n,2*k-1)*a(n-2*k+1). - Ralf Stephan, Apr 29 2004
a(n) ~ (sqrt(2)/2)*n!/log(1+sqrt(2))^(n+1). - Conjectured by Simon Plouffe, Feb 17 2007.
From Andrew Hone, Feb 22 2007: (Start)
This formula can be proved using the techniques in the article by Philippe Flajolet (see links) [see Theorem 5 and Table 2, noting that 1/(1-sinh(x)) just has a simple pole at x=log(1+sqrt(2))]. (End)
a(n) = Sum_{k=1..n} Sum_{i=0..k} (-1)^i*(k-2*i)^n*binomial(k,i)/2^k, n > 0, a(0)=1. - Vladimir Kruchinin, May 28 2011
Row sums (apart from a(0)) of A196776. - Peter Bala, Oct 06 2011
Row sums of A193474. - Peter Luschny, Oct 07 2011
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A003724 and A000111. - Peter Bala, Dec 06 2011
From Sergei N. Gladkovskii, Jun 01 2012: (Start)
Let E(x) be the e.g.f., then
E(x) = -1/x + 1/(x*(1-x))+ x^3/((1-x)*((1-x)*G(0) - x^2)); G(k) = (2*k+2)*(2*k+3)+x^2-(2*k+2)*(2*k+3)*x^2/G(k+1); (continued fraction).
E(x) = -1/x + 1/(x*(1-x))+ x^3/((1-x)*((1-x)*G(0) - x^2)); G(k) = 8*k+6+x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)); (continued fraction).
E(x) = 1/(1 - x*G(0)); G(k) = 1 + x^2/(2*(2*k+1)*(4*k+3) + 2*x^2*(2*k+1)*(4*k+3)/(-x^2 - 4*(k+1)*(4*k+5)/G(k+1))); (continued fraction).
(End).
E.g.f. 1/(1 - x*G(0)) where G(k) = 1 - x^2/( (2*k+1)*(2*k+3) - (2*k+1)*(2*k+3)^2/(2*k+3 - (2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 01 2012
O.g.f A(x) satisfies A(x) = 1 + ( A(x/(1-x))/(1-x) - A(x/(1+x))/(1+x) )/2. - Paul D. Hanna, Aug 19 2024

More terms from Christian G. Bower, Oct 15 1999

A005046 Number of partitions of a 2n-set into even blocks.

Original entry on oeis.org

1, 1, 4, 31, 379, 6556, 150349, 4373461, 156297964, 6698486371, 337789490599, 19738202807236, 1319703681935929, 99896787342523081, 8484301665702298804, 802221679220975886631, 83877585692383961052499, 9640193854278691671399436, 1211499609050804749310115589

Offset: 0

N. J. A. Sloane

Conjecture: Taking the sequence modulo an integer k gives an eventually periodic sequence. For example, the sequence taken modulo 10 is [1, 1, 4, 1, 9, 6, 9, 1, 4, 1, 9, 6, 9, 1, 4, 1, 9, 6, 9, ...], with an apparent period [1, 4, 1, 9, 6, 9] beginning at a(1), of length 6. Cf. A006154. - Peter Bala, Apr 12 2023

Louis Comtet, Analyse Combinatoire Tome II, pages 61-62.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 225, 3rd line of table.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 51 terms from T. D. Noe)
C. Ahmed, P. Martin, and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015-2019.
Steven R. Finch, Moments of sums, April 23, 2004 [Cached copy, with permission of the author]
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
J. Riordan, Letter, Jul 06 1978
J. Shallit, Letter to N. J. A. Sloane, Jan 13 1976.
Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.

See A156289 for the table of partitions of a 2n-set into k even blocks.
For partitions into odd blocks see A003724 and A136630.
Cf. A000110, A003724.

a:= proc(n) option remember;
      `if`(n=0, 1, add(binomial(2*n-1, 2*k-1) *a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 12 2011
# second Maple program:
a := n -> add(binomial(2*n,k)*(-1)^k*BellB(k,1/2)*BellB(2*n-k,1/2), k=0..2*n):
seq(a(n), n=0..18); # after Emanuele Munarini,_Peter Luschny_, Sep 10 2017
B := BellMatrix(n -> modp(n, 2), 31): # defined in A264428.
seq(add(k, k in B[2*n + 1]),n=0..15); # Peter Luschny, Aug 13 2019

NestList[ Factor[ D[#, {x, 2}]] &, Exp[ Cosh[x] - 1], 16] /. x -> 0
a[0] = 1; a[n_] := Sum[Sum[(i-k)^(2*n)*Binomial[2*k, i]*(-1)^i, {i, 0, k-1}]/(2^(k-1)*k!), {k, 1, 2*n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 07 2015, after Vladimir Kruchinin *)
Table[Sum[BellY[2 n, k, 1 - Mod[Range[2 n], 2]], {k, 0, 2 n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
With[{nn=40},Abs[Take[CoefficientList[Series[Exp[Cos[x]-1],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]]] (* Harvey P. Dale, Feb 06 2017 *)

a(n):= sum(1/k!*sum(binomial(k,m)/(2^(m-1))*sum(binomial(m,j) *(2*j-m)^(2*n), j,0,m/2)*(-1)^(k-m), m,0,k), k,1,2*n); /* Vladimir Kruchinin, Aug 05 2010 */

a(n):=sum(sum((i-k)^(2*n)*binomial(2*k,i)*(-1)^(i),i,0,k-1)/(2^(k-1)*k!),k,1,2*n); /* Vladimir Kruchinin, Oct 04 2012 */

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def a(n): return 1 if n==0 else sum(binomial(2*n - 1, 2*k - 1)*a(n - k) for k in range(1, n + 1))
print([a(n) for n in range(21)]) # Indranil Ghosh, Sep 11 2017, after Maple program by Alois P. Heinz

E.g.f.: exp(cosh(x) - 1) (or exp(cos(x)-1) ).
a(n) = Sum_{k=1..n} binomial(2*n-1, 2*k-1)*a(n-k). - Vladeta Jovovic, Apr 10 2003
a(n) = sum(1/k!*sum(binomial(k,m)/(2^(m-1))*sum(binomial(m,j)*(2*j-m)^(2*n),j,0,m/2)*(-1)^(k-m),m,0,k),k,1,2*n), n>0. - Vladimir Kruchinin, Aug 05 2010
a(n) = Sum_{k=1..2*n} Sum_{i=0..k-1} ((i-k)^(2*n)*binomial(2*k,i)*(-1)^i)/(2^(k-1)*k!), n>0, a(0)=1. - Vladimir Kruchinin, Oct 04 2012
E.g.f.: E(0)-1, where E(k) = 2 + (cosh(x)-1)/(2*k + 1 - (cosh(x)-1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
a(n) = Sum_{k=0..2*n} binomial(2*n,k)*(-1)^k*S_k(1/2)*S_{2*n-k}( 1/2), where S_n(x) is the n-th Bell polynomial (or exponential polynomial). - Emanuele Munarini, Sep 10 2017

A002017 Expansion of e.g.f. exp(sin(x)).

Original entry on oeis.org

1, 1, 1, 0, -3, -8, -3, 56, 217, 64, -2951, -12672, 5973, 309376, 1237173, -2917888, -52635599, -163782656, 1126610929, 12716052480, 20058390573, -495644917760, -3920482183827, 4004259037184, 256734635981833, 1359174582304768

Offset: 0

N. J. A. Sloane

Number of set partitions of 1..n into odd parts with an even number of parts of size == 3 (mod 4), minus the number of such partitions with an odd number of parts of size == 3 (mod 4). - Franklin T. Adams-Watters, Apr 29 2010

			For n=6, there are 6 partitions with part sizes [5,1], 10 with sizes [3^2], 20 with sizes [3,1^3], and 1 with sizes [1^6]; 6 + 10 - 20 + 1 = -3. - _Franklin T. Adams-Watters_, Apr 29 2010

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
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).

T. D. Noe, Table of n, a(n) for n=0..100
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

a(2n) = A007301(n), |a(2n+1)| = |A003722(n)|.
Cf. A003724. - Franklin T. Adams-Watters, Apr 29 2010
Cf. A047687, A047688 for numerators & denominators of the series of exp(sin(x)) at x = 0.

max = 25; se = Series[Exp[Sin[x]], {x, 0, max}]; CoefficientList[se, x] *Range[0, max]! (* Jean-François Alcover, Jun 26 2013 *)

a(n):=2*sum((sum((2*i-n+2*j)^n*binomial(n-2*j,i)*(-1)^(n-j-i),i,0,(n-2*j)/2))/(2^(n-2*j)*(n-2*j)!),j,0,(n-1)/2); /* Vladimir Kruchinin, Jun 10 2011 */

a(n):=if n=0 then 1 else (n-1)!*sum((-1)^(k)/(2*k)!*a(n-2*k-1)/(n-2*k-1)!,k,0,(n-1)/2); /* Vladimir Kruchinin, Feb 25 2015 */

my(x='x+O('x^33)); Vec(serlaplace(exp(sin(x)))) \\ Joerg Arndt, Apr 01 2017

a(n) = 2*Sum_{j=0..(n-1)/2} Sum_{i=0..(n-2*j)/2} (2*i-n+2*j)^n*C(n-2*j,i)*(-1)^(n-j-i)/(2^(n-2*j)*(n-2*j)!), n>0, a(0)=1. - Vladimir Kruchinin, Jun 10 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1-x^2)*d/dx. Cf. A003724. - Peter Bala, Dec 06 2011
E.g.f.: 1 + sin(x)/T(0), where T(k) = 4*k+1 - sin(x)/(2 + sin(x)/(4*k+3 - sin(x)/(2 + sin(x)/T(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
E.g.f.: 2/Q(0), where Q(k) = 1 + 1/( 1 - sin(x)/( sin(x) - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013
E.g.f.: E(0)-1, where E(k) = 2 + sin(x)/(2*k + 1 - sin(x)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
a(n) = (n-1)!*Sum_{k=0..(n-1)/2} ((-1)^k/(2*k)!)*a(n-2*k-1)/(n-2*k-1)!, a(0)=1. - Vladimir Kruchinin, Feb 25 2015

Extended with signs by Christian G. Bower, Nov 15 1998

A050330 Number of factorizations of n into numbers with an odd number of prime factors.

Original entry on oeis.org

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

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

R. J. Mathar, Table of n, a(n) for n = 1..2519

Cf. A001055, A026424, A050331. a(p^k)=A000009. a(A002110)=A003724.

Dirichlet g.f.: Product_{n in A026424} (1/(1-1/n^s)).

a(n) = A050331(A101296(n)). - R. J. Mathar, May 26 2017

A096579 Number of partitions of an n-set with exactly one even block.

Original entry on oeis.org

0, 1, 3, 7, 25, 91, 329, 1415, 6297, 29431, 151085, 802099, 4506957, 26836083, 165586321, 1074740079, 7268876881, 50985776815, 372854157589, 2820244541675, 22087612114805, 179014336044171, 1495539626297689, 12894921568568999, 114481871464864825

Offset: 1

Vladeta Jovovic, Aug 13 2004

Alois P. Heinz, Table of n, a(n) for n = 1..591

Cf. A003724.

b:= proc(n, t) option remember; `if`(n=0, t, add(
      `if`(t=1 and j::even, 0, binomial(n-1, j-1)*
       b(n-j, `if`(j::even, 1, t))), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, May 10 2016

Drop[ Range[0, 24]! CoefficientList[ Series[ E^Sinh[x]*(Cosh[x] - 1), {x, 0, 24}], x], 1] (* Robert G. Wilson v, Aug 17 2004 *)

E.g.f.: exp(sinh(x))*(cosh(x)-1). More generally, e.g.f. for the number of partitions of n-set with exactly k even blocks is 1/k!*exp(sinh(x))*(cosh(x)-1)^k.

More terms from Robert G. Wilson v, Aug 17 2004

A306347 Expansion of e.g.f. exp((sin(x) + sinh(x))/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 22, 57, 128, 389, 1904, 9329, 38040, 132147, 542648, 3283633, 20997824, 114657097, 536178880, 2784500161, 19876061312, 153326461311, 1034551839872, 6051063485481, 38079448046208, 312420426154893, 2785055242928768, 22141255520251313

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

Number of partitions of n-set into blocks congruent to 1 mod 4.

Seiichi Manyama, Table of n, a(n) for n = 0..608

Cf. A002017, A003724, A005046, A013025, A016813, A035451, A291975, A307978, A307979, A351969.

nmax = 28; CoefficientList[Series[Exp[(Sin[x] + Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 28}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sin(x)+sinh(x))/2))) \\ Seiichi Manyama, Mar 17 2022

a(n) = if(n==0, 1, sum(k=0, (n-1)\4, binomial(n-1, 4*k)*a(n-4*k-1))); \\ Seiichi Manyama, Mar 17 2022

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * a(n-4*k-1). - Seiichi Manyama, Mar 17 2022

A136632 a(n) = Sum_{k=0..n} A136630(n,k) * 2^(nk).

Original entry on oeis.org

1, 2, 16, 520, 66560, 33882144, 69055086592, 564152735105152, 18462508115518554112, 2418626436468567646929408, 1267795674038260517176495570944, 2658560573512321601282555747644737536

Offset: 0

Paul D. Hanna, Jan 14 2008

nonn

			From _Paul D. Hanna_, Nov 25 2009: (Start)
E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 520*x^3/3! + 66560*x^4/4! +...
A(x) = 1 + sinh(2*x) + sinh(4*x)^2/2! + sinh(8*x)^3/3! + sinh(16*x)^4/4! +...+ sinh(2^n*x)^n/n! +...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(sinh(x)):
G(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 12*x^5/5! + 37*x^6/6! +...+ A003724(n)*x^n/n! +... (End)

Robert Israel, Table of n, a(n) for n = 0..57

Cf. A136630, A003724 (row sums of A136630).

Cf. A003724 (exp(sinh x)). [From Paul D. Hanna, Nov 25 2009]

N:= 20: # to get a(0)..a(N)
E:= add(sinh(2^n*x)^n/n!,n=0..N):
S:= series(E,x,N+1):
seq(coeff(S,x,j)*j!,j=0..N); # Robert Israel, Jan 17 2018

{a(n)=sum(k=0,n,2^(n*k)*polcoeff(x^k/prod(j=0,k\2,1-(2*j+k-2*(k\2))^2*x^2 +x*O(x^n)),n))}

{a(n)=n!*polcoeff(sum(k=0,n,sinh(2^k*x +x*O(x^n))^k/k!),n)} \\ Paul D. Hanna, Nov 25 2009

{a(n)=n!*polcoeff(exp(2^n*sinh(x +x*O(x^n))),n)} \\ Paul D. Hanna, Nov 25 2009

From Paul D. Hanna, Nov 25 2009: (Start)
E.g.f.: Sum_{n>=0} sinh(2^n*x)^n/n!.
a(n) = [x^n/n! ] exp(2^n*sinh(x)).
(End)

A032310 Number of ways to partition n labeled elements into sets of odd sizes, with all sizes different.

Original entry on oeis.org

1, 1, 0, 1, 4, 1, 6, 1, 64, 505, 130, 1321, 1024, 13157, 2380, 395851, 5782144, 1639617, 24545706, 16100905, 306621184, 292018525, 6304002100, 1549052715, 507969498304, 11794047630801, 3164830777316, 75389026652551, 48756350408224, 1240389053007865

Offset: 0

Christian G. Bower

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500
C. G. Bower, Transforms (2)

Cf. A003724.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-2), j=0..min(1, n/i))))
    end:
a:= n-> b(n, iquo(n+1, 2)*2-1):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 08 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n - i*j, i-2], {j, 0, Min[1, n/i]}]]]; a[n_] := b[n, Quotient[n+1, 2]*2-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

"EGJ" (unordered, element, labeled) transform of 1, 0, 1, 0... (odds)

E.g.f.: Product_{k>0} (1+x^(2*k-1)/(2*k-1)!). - Vladeta Jovovic, Jan 16 2004

Description corrected by Vladeta Jovovic, Aug 18 2004

a(0)=1 prepended by Alois P. Heinz, Mar 08 2015

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.

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A136630 Triangular array: T(n,k) counts the partitions of the set [n] into k odd sized blocks.

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A006154 Number of labeled ordered partitions of an n-set into odd parts.

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A005046 Number of partitions of a 2n-set into even blocks.

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A002017 Expansion of e.g.f. exp(sin(x)).

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