A001935 -id:A001935 - 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

A000726 Number of partitions of n in which no parts are multiples of 3.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 57, 70, 89, 108, 135, 163, 202, 243, 297, 355, 431, 513, 617, 731, 874, 1031, 1225, 1439, 1701, 1991, 2341, 2731, 3197, 3717, 4333, 5022, 5834, 6741, 7803, 8991, 10375, 11923, 13716, 15723, 18038, 20628, 23603

Offset: 0

N. J. A. Sloane

Case k=4, i=3 of Gordon Theorem.
Expansion of q^(-1/12)*eta(q^3)/eta(q) in powers of q. - Michael Somos, Apr 20 2004
Euler transform of period 3 sequence [1,1,0,...]. - Michael Somos, Apr 20 2004
Also the number of partitions with at most 2 parts of size 1 and all differences between parts at distance 3 are greater than 1. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,2] (for example, [2,2,1,1] does not qualify because the difference between the first and the fourth parts is equal to 1). - Emeric Deutsch, Apr 18 2006
Also the number of partitions of n where no part appears more than twice. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,1,1]. - Emeric Deutsch, Apr 18 2006
Also the number of partitions of n with least part either 1 or 2 and with differences of consecutive parts at most 2. Example: a(6)=7 because we have [4,2], [3,2,1], [3,1,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1] and [1,1,1,1,1,1]. - Emeric Deutsch, Apr 18 2006
Equals left border of triangle A174714. - Gary W. Adamson, Mar 27 2010
Triangle A113685 is equivalent to p(x) = p(x^2) * A000009(x); given A000041(x) = p(x). Triangle A176202 is equivalent to p(x) = p(x^3) * A000726(x). - Gary W. Adamson, Apr 11 2010
Convolution of A035382 and A035386. - Vaclav Kotesovec, Aug 23 2015
The number of partitions of n in which no parts are multiples of k equals the number of partitions of n where no part appears more than k-1 times. - Gregory L. Simay, Oct 15 2022

			There are a(6)=7 partitions of 6 into parts != 0 (mod 3):
[ 1]  [5,1],
[ 2]  [4,2],
[ 3]  [4,1,1],
[ 4]  [2,2,2],
[ 5]  [2,2,1,1],
[ 6]  [2,1,1,1,1], and
[ 7]  [1,1,1,1,1,1]
.
From _Joerg Arndt_, Dec 29 2012: (Start)
There are a(10)=22 partitions p(1)+p(2)+...+p(m)=10 such that p(k)!=p(k-2) (that is, no part appears more than twice):
[ 1]  [ 3 3 2 1 1 ]
[ 2]  [ 3 3 2 2 ]
[ 3]  [ 4 2 2 1 1 ]
[ 4]  [ 4 3 2 1 ]
[ 5]  [ 4 3 3 ]
[ 6]  [ 4 4 1 1 ]
[ 7]  [ 4 4 2 ]
[ 8]  [ 5 2 2 1 ]
[ 9]  [ 5 3 1 1 ]
[10]  [ 5 3 2 ]
[11]  [ 5 4 1 ]
[12]  [ 5 5 ]
[13]  [ 6 2 1 1 ]
[14]  [ 6 2 2 ]
[15]  [ 6 3 1 ]
[16]  [ 6 4 ]
[17]  [ 7 2 1 ]
[18]  [ 7 3 ]
[19]  [ 8 1 1 ]
[20]  [ 8 2 ]
[21]  [ 9 1 ]
[22]  [ 10 ]
(End)

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145.
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 and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See p. 79.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
Eric Weisstein's World of Mathematics, Partition function b_k.
Wikipedia, Glaisher's Theorem.

Cf. A000009 (no multiples of 2), A001935 (no of 4), A035959 (no of 5), A219601 (no of 6), A035985, A001651, A003105, A035361, A035360.
Cf. A174714. - Gary W. Adamson, Mar 27 2010
Cf. A113685, A176202. - Gary W. Adamson, Apr 11 2010
Cf. A046913.
Column k=3 of A286653.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a000726 n = p a001651_list n where
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 23 2011

g:=product(1+x^j+x^(2*j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..50); # Emeric Deutsch, Apr 18 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(irem(d, 3)=0, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 17 2017

f[0] = 1; f[n_] := Coefficient[Expand@ Product[1 + x^k + x^(2k), {k, n}], x^n]; Table[f@n, {n, 0, 40}] (* Robert G. Wilson v, Nov 10 2006 *)
QP = QPochhammer; CoefficientList[QP[q^3]/QP[q] + O[q]^60, q] (* Jean-François Alcover, Nov 24 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 02 2016 *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 3], 0, 2] ], {n, 0, 50}] (* Robert Price, Jul 28 2020 *)
Table[Count[IntegerPartitions[n],?(NoneTrue[Mod[#,3]==0&])],{n,0,50}] (* _Harvey P. Dale, Sep 06 2022 *)

a(n)=if(n<0,0,polcoeff(eta(x^3+x*O(x^n))/eta(x+x*O(x^n)),n))

lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q))} \\ Altug Alkan, Mar 20 2018

G.f.: 1/(Product_{k>=1} (1-x^(3*k-1))*(1-x^(3*k-2))) = Product_{k>=1} (1 + x^k + x^(2*k)) (where 1 + x + x^2 is the 3rd cyclotomic polynomial).
a(n) = A061197(n, n).
Given g.f. A(x) then B(x) = x*A(x^6)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u,v,w) = +v^2 +v*w^2 -v*u^2 +3*u^2*w^2. - Michael Somos, May 28 2006
G.f.: P(x^3)/P(x) where P(x) = Product_{k>=1} (1 - x^k). - Joerg Arndt, Jun 21 2011
a(n) ~ 2*Pi * BesselI(1, sqrt((12*n + 1)/3)*Pi/3) / (3*sqrt(12*n + 1)) ~ exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)) * (1 + (Pi/36 - 9/(16*Pi))/sqrt(n) + (Pi^2/2592 - 135/(512*Pi^2) - 5/64)/n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 13 2017
a(n) = (1/n)*Sum_{k=1..n} A046913(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^(3*k)))). - Ilya Gutkovskiy, Aug 15 2018

More terms from Olivier Gérard

A042968 Numbers not divisible by 4.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 1

N. J. A. Sloane, Dec 11 1999

Equivalently, numbers whose square part is odd. Cf. A028982. - Peter Munn, Jul 14 2020
More generally the sequence of numbers not divisible by some fixed integer m >= 2 is given by a(n,m) = 1 + n + floor(n/(m-1)). - Benoit Cloitre, Jul 11 2009
Also a(n,m) = floor((m*n-1)/(m-1)) [with offset 1]. - Gary Detlefs, May 14 2011
Numbers not having more even than odd divisors: A048272(a(n)) >= 0. - Reinhard Zumkeller, Jan 21 2012
Extending the comments of Benoit Cloitre (Jul 11 2009) and Gary Detlefs (May 14 2011), the g.f. is A(m,x) = (1-x^m) / ((1-x^(m-1))*(1-x)^2) where m >= 2 is fixed. - Werner Schulte, Apr 26 2018

			G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 6*x^4 + 7*x^5 + 9*x^6 + 10*x^7 + 11*x^8 + ... - _Michael Somos_, Jun 17 2018

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A001651, A001935, A028982, A070048, A042965, A064680.
Cf. A071619 (partial sums); A008586 (complement).
Numbers that are congruent to {k0,k1,k2} mod 4: A004772, A004773, A042965, a(n).

a042968 = (`div` 3) . (subtract 1) . (* 4)
a042968_list = filter ((/= 0) . (`mod` 4)) [1..]
-- Reinhard Zumkeller, Sep 02 2012

[n+1+Floor(n/3): n in [0..80]]; // Vincenzo Librandi, Aug 03 2015

seq(n+floor((n-1)/3), n=1..80); # Muniru A Asiru, Feb 17 2019

Select[Table[n,{n,200}], Mod[#,4] != 0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
LinearRecurrence[{1,0,1,-1},{1,2,3,5},80]  (* or *) Drop[Range[110],{4,-1,4}] (* Harvey P. Dale, Jan 07 2023 *)

PARI
```
{a(n) = 1 + n + n\3};
```

def A042968(n): return n+(n-1)//3 # Chai Wah Wu, Apr 13 2025

[1+n+floor(n/3) for n in (0..80)] # G. C. Greubel, Feb 17 2019

a(n) = a(n-1) + a(n-3) - a(n-4).
a(n) = a(n-3) + 4, with a(1) = 1.
G.f.: x * (1+x) * (1+x^2) / ( (1+x+x^2)*(1-x)^2 ). - Michael Somos, Jan 12 2000
A064680(A064680(a(n))) = a(n). - Reinhard Zumkeller, Oct 19 2001
Nearest integer to (Sum_{k>n} 1/k^4)/(Sum_{k>n} 1/k^5). - Benoit Cloitre, Jun 12 2003
a(n) = n + 1 + floor(n/3). - Benoit Cloitre, Jul 11 2009
a(n) = floor((4*n+3)/3). - Gary Detlefs, May 14 2011
A214546(a(n)) >= 0 for n > 0. - Reinhard Zumkeller, Jul 20 2012
a(n) = 2*n - ceiling(2*n/3) + 1. - Arkadiusz Wesolowski, Sep 21 2012
Sum_{k=0..n} a(n) = A071619(n+1). - L. Edson Jeffery, Jul 30 2014
The g.f. A(x) satisfies x*A(x)^2 = (B(x)/x)^2 + (B(x)/x), where B(x) is the o.g.f. of A042965. - Peter Bala, Apr 12 2017
a(n) = (12*n + 6 + 3*cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017
Euler transform of length 4 sequence [2, 0, 1, -1]. - Michael Somos, Jun 17 2018
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Jun 17 2018
E.g.f.: (2/3)*exp(x)*(1 + 2*x) + (1/9)*exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)). - Stefano Spezia, Nov 16 2019
a(n) = (12*n + 6 + w^(2*n)*(w + 2) - w^n*(w - 1))/9 where w = (-1 + sqrt(-3))/2. - Guenther Schrack, Jun 07 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/8. - Amiram Eldar, Dec 05 2021

Edited by Peter Munn, Nov 16 2019

I restored my original (1999) definition and offset, which in the intervening 21 years had been lost. - N. J. A. Sloane, Jun 12 2021

A035959 Number of partitions of n in which no parts are multiples of 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, 127, 164, 205, 262, 325, 409, 505, 628, 769, 950, 1156, 1414, 1713, 2081, 2505, 3026, 3625, 4352, 5192, 6200, 7364, 8756, 10357, 12258, 14450, 17034, 20006, 23500, 27510, 32200, 37582, 43846, 51022

Offset: 0

Olivier Gérard

Also number of partitions with at most 4 parts of size 1 and differences between parts at distance 6 are greater than 1.
Also number of partitions of n where no part appears more than four times.
Case k=7, i=5 of Gordon Theorem.

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 13*x^7 + 19*x^8 + ...
G.f. = q + q^7 + 2*q^13 + 3*q^19 + 5*q^25 + 6*q^31 + 10*q^37 + 13*q^43 + ...
a(6) counts these partitions: 6, 42, 411, 33, 321, 3111, 2211, 21111, 111111. - _Clark Kimberling_, Mar 09 2014

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
G. N. Watson, Ramanujans Vermutung ueber Zerfaellungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression Y = C/B in the notation of p. 106. [Added by _N. J. A. Sloane_, Nov 13 2009]
Eric Weisstein's World of Mathematics, Partition Function b_k.
Wikipedia, Glaisher's Theorem

Cf. A061198, A061199, A047201.
Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10).
Cf. A096938, A133563, A320607.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a035959 = p a047201_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Dec 17 2011

max = 47; f[x_] := (x^5-1)/(x-1); g[x_] := Product[f[x^k], {k, 1, max}]; CoefficientList[ Series[g[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 29 2011, after Michael Somos *)
t = Flatten[Table[5 n + r, {n, 0, 60}, {r, 1, 4}]]; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 8}] (* shows partitions *)
a[n_] := Length@p@n; a /@ Range[0, 50] (* Clark Kimberling, Mar 09 2014 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^5]/QP[q] + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, after Michael Somos *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 5], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)
Table[Count[IntegerPartitions[n],?(NoneTrue[Mod[#,5]==0&])],{n,0,50}] (* _Harvey P. Dale, Dec 25 2021 *)

{a(n) = if( n<0, 0, polcoeff( eta(x^5 + x * O(x^n)) / eta(x + x * O(x^n)), n))}; /* Michael Somos, May 28 2006 */

G.f.: Product_{j>=1} (1 + x^j + x^2j + x^3j + x^4j). - Jon Perry, Mar 30 2004
G.f.: Product_{n>0, n==1, 2, 3, 4 mod 5} 1/(1-q^n).
Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - u*v - 5*u^2*v^2. - Michael Somos, May 28 2006
Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v + 5*v^2*(u + w) - (u^2 + u*w + w^2). - Michael Somos, May 28 2006
Euler transform of period 5 sequence [ 1, 1, 1, 1, 0, ...]. - Michael Somos, May 28 2006
G.f.: Product_{k > 0} P5(x^k) where P5 is 5th cyclotomic polynomial.
Convolution inverse is A145466. - Michael Somos, Jun 26 2014
a(n) ~ 2*Pi * BesselI(1, 2*sqrt((6*n + 1)/5) * Pi/3) / (5*sqrt(6*n + 1)) ~ exp(2*Pi*sqrt(2*n/15)) / (3^(1/4) * 10^(3/4) * n^(3/4)) * (1 + (Pi/(3*sqrt(15)) - 3*sqrt(15)/(16*Pi)) / sqrt(2*n) + (Pi^2/540 - 225/(1024*Pi^2) - 5/32) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A116073(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + x^(3*k))/(k*(1 - x^(5*k)))). - Ilya Gutkovskiy, Aug 15 2018

A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.

Original entry on oeis.org

1, 2, 5, 10, 18, 32, 55, 90, 144, 226, 346, 522, 777, 1138, 1648, 2362, 3348, 4704, 6554, 9056, 12425, 16932, 22922, 30848, 41282, 54946, 72768, 95914, 125842, 164402, 213901, 277204, 357904, 460448, 590330, 754368, 960948, 1220370, 1545306

Offset: 0

N. J. A. Sloane

The Cayley reference is actually to A079006. - Michael Somos, Feb 24 2011
In the math overflow link is a conjecture that a(n) == a(9*n + 2) (mod 4).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of 4-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019
The g.f. in the form A(x) = Sum_{k >= 0} x^(k*(k+1)) / (1 + 2*Sum_{k >= 1} (-1)^k * x^(k^2)) == Sum_{k >= 0} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + 90*x^7 + 144*x^8 + ...
G.f. = q + 2*q^5 + 5*q^9 + 10*q^13 + 18*q^17 + 32*q^21 + 55*q^25 + 90*q^29 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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..1000
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
joro on Math Overflow, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?.
T. Kathiravan and S. N. Fathima, On L-regular bipartitions modulo L, The Ramanujan Journal 44.3 (2017): 549-558.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 07, 1976
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001935, A022568, A079006, A082304, A098613, A127391, A127392, A210656.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548, A333374.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> [2,2,2,0] [modp(n-1,4)+1]): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
f:=(k,M) -> mul(1-q^(k*j),j=1..M); LRBP := (L,M) -> (f(L,M)/f(1,M))^2; S := L -> seriestolist(series(LRBP(L,80),q,60)); S(4); # N. J. A. Sloane, Oct 20 2019

m = 38; CoefficientList[ Series[ Product[ (1 - x^(4*k))/(1 - x^k), {k, 1, m}]^2 , {x, 0, m}], x] (* Jean-François Alcover, Sep 02 2011, after g.f. *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]) / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^2, x^2])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)

{a(n) = if( n<0, 0, polcoeff( (eta(x^4 + x * O(x^n)) / eta(x + x * O(x^n)))^2, n))};

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 / if(k%4, 1 - x^k, 1), 1 + x * O(x^n))^2, n))};

G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....
Convolution square of A001935. A079006(n) = (-1)^n a(n).
Expansion of q^(-1/4) * (1/2) * (k / k')^(1/2) in powers of q.
Euler transform of period 4 sequence [ 2, 2, 2, 0, ...].
Given g.f. A(x), then B(q) = (q * A(q^4))^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + 16*u) * (1 + 16*v) * v - u^2. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (1 + 4*u*v)^2. - Michael Somos, Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - x^k))^2.
Equals A000009 convolved with A098613. - Gary W. Adamson, Mar 24 2011
a(9*n + 2) = a(n) + 4 * A210656(3*n). - Michael Somos, Apr 02 2012
Convolution inverse is A082304. - Michael Somos, May 16 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082304. - Michael Somos, May 16 2015
Expansion of f(-x^4)^2 / f(-x)^2 = psi(x^2) / phi(-x) = psi(-x)^2 / phi(-x)^2 = psi(x)^2 / phi(-x^2)^2 = psi(x^2)^2 / psi(-x)^2 = chi(x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x)^4) = 1 / (chi(-x)^2 * chi(-x^2)^2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, May 16 2015
a(n) ~ exp(Pi*sqrt(n)) / (8*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 18 2015
G.f.: A(x) = Sum_{n >= 0} x^(n*(n+1)) / Sum_{n = -oo..oo} (-1)^n*x^(n^2). - Peter Bala, Feb 19 2021

cp's OEIS Frontend

A174715 Triangle read by rows, Q*M as infinite lower triangular matrices. Q = A001935 shifted down four times by columns, M = A174712.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A014605 Partial sums of A001935; at one time this was conjectured to agree with A007478.

Views

Author

Keywords

Links

Programs

Maple

Mathematica

Formula

A115247 2^a(n) divides A001935(n) but 2^(a(n)+1) does not.

Views

Author

Keywords

Comments

Links

Crossrefs

A115248 Values such that A115247(a(n))=1. Values such that A001935(a(n))==2 (mod 4).

Views

Author

Keywords

Comments

Links

Crossrefs

A340508 Let ped(n) denote the number of partitions of n in which the even parts are distinct (A001935); a(n) = ped(9*n+7).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Julia

Magma

Maple

Mathematica

Maxima

Maxima

PARI

PARI

PARI

PARI

Python

Python

SageMath

Formula

A000726 Number of partitions of n in which no parts are multiples of 3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell