A328545 -id:A328545 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, 105, 132, 166, 208, 258, 320, 395, 484, 592, 722, 876, 1060, 1280, 1539, 1846, 2210, 2636, 3138, 3728, 4416, 5222, 6163, 7256, 8528, 10006, 11716, 13696, 15986, 18624, 21666, 25169, 29190, 33808, 39104, 45164

Offset: 0

N. J. A. Sloane, Simon Plouffe, Robert G. Wilson v

Also number of partitions of n where no part appears more than three times.
a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n is in A062717 (see Fink et al.).
Also number of partitions of n in which the least part and the differences between consecutive parts is at most 3. Example: a(5)=6 because we have [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1]. - Emeric Deutsch, Apr 19 2006
Equals A000009 convolved with its aerated variant, = polcoeff A000009 * A000041 * A010054 (with alternate signs). - Gary W. Adamson, Mar 16 2010
Equals left border of triangle A174715. - Gary W. Adamson, Mar 27 2010
The Cayley reference is actually to A083365. - Michael Somos, Feb 24 2011
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution of A000009 and A035457. - Vaclav Kotesovec, Aug 23 2015
Convolution inverse is A082303. - Michael Somos, Sep 30 2017
The g.f. in the form Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1-x^k) = Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1+x^k-2*x^k) == Sum_{n >= 0} x^(n*(n+1)/2) (mod 2). It follows that a(n) is odd iff n = k*(k + 1)/2 for some nonnegative integer k. Cf. A333374. - Peter Bala, Jan 08 2025

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 16*x^8 + 22*x^9 + ...
G.f. = q + q^9 + 2*q^17 + 3*q^25 + 4*q^33 + 6*q^41 + 9*q^49 + 12*q^57 + 16*q^65 + 22*q^73 + ...
a(5)=6 because we have [5], [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1].

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.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.2).
M. D. Hirschhorn, The Power of q, Springer, 2017. See ped page 303ff.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 241.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 9.)
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 Theorem 1.4 p. 75.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Cristina Ballantine and Mircea Merca, Parity of sums of partition numbers and squares in arithmetic progressions, The Ramanujan Journal, 2016.
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.]
S.-C. Chen, On the number of partitions with distinct even parts, Discrete Math., 311 (2011), 940-943.
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
M. D. Hirschhorn and J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
Joro, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?
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.
Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
Alexander Patkowski, On some partitions where even parts do not repeat, Demonstratio Mathematica Volume 42, Issue 2 (Jun 2009), pp. 259-263.
Eric Weisstein's World of Mathematics, Partition Function b_k and Partition Function P.
Wikipedia, Glaisher's Theorem.

Cf. A000009, A000726, A001936, A035959, A035985, A042968, A061198, A061199, A070048, A081055, A081056, A083365, A098491, A098492, A219601.
Cf. A000041, A010054. - Gary W. Adamson, Mar 16 2010
Cf. A174715. - Gary W. Adamson, Mar 27 2010
Cf. A082303.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a001935 = p a042968_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, Sep 02 2012

g:=product((1+x^j)*(1+x^(2*j)),j=1..50): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..48); # Emeric Deutsch, Apr 19 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(irem(d, 4)=0, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 24 2015

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8), {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 08 2011 *)
CoefficientList[Series[Product[1+x^j+x^(2j)+x^(3j), {j,1,48}], {x,0,48}],x] (* Jean-François Alcover, May 26 2011, after Jon Perry *)
QP = QPochhammer; CoefficientList[QP[q^4]/QP[q] + O[q]^50, q] (* Jean-François Alcover, Nov 24 2015 *)
a[0] = 1; a[n_] := a[n] = Sum[a[n-j] DivisorSum[j, If[Divisible[#, 4], 0, #]&], {j, 1, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 4], 0, 2] ], {n, 0, 49}] (* Robert Price, Jul 28 2020 *)

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

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 8*n + 1) - 1)\2, prod(i=1, k, (1 + x^i) / (x^-i - 1), 1 + x * O(x^n))), n))}; /* Michael Somos, Jun 01 2004 */

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(1+(-x)^m+x*O(x^n))/m)),n)} \\ Paul D. Hanna, Jul 24 2013

Euler transform of period 4 sequence [ 1, 1, 1, 0, ...].
Expansion of q^(-1/8) * eta(q^4) / eta(q) in powers of q. - Michael Somos, Mar 19 2004
Expansion of psi(-x) / phi(-x) = psi(x) / phi(-x^2) = psi(x^2) / psi(-x) = chi(x) / chi(-x^2)^2 = 1 / (chi(x) * chi(-x)^2) = 1 / (chi(-x) * chi(-x^2)) = f(-x^4) / f(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 08 2011
G.f.: Product(j>=1, 1 + x^j + x^(2*j) + x^(3*j)). - Jon Perry, Mar 30 2004
G.f.: Product_{k>=1} (1+x^k)^(2-k%2). - Jon Perry, May 05 2005
G.f.: Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)) = 1 + Sum_{k>0}(Product_{i=1..k} (x^i + 1) / (x^-i - 1)).
G.f.: Sum_{n>=0} ( x^(n*(n+1)/2) * Product_{k=1..n} (1+x^k)/(1-x^k) ). - Joerg Arndt, Apr 07 2011
G.f.: P(x^4)/P(x) where P(x) = Product_{k>=1} 1-x^k. - Joerg Arndt, Jun 21 2011
A083365(n) = (-1)^n a(n). Convolution square is A001936. a(n) = A098491(n) + A098492(n). a(2*n) = A081055(n). a(2*n + 1) = A081056(n).
G.f.: (1+ 1/G(0))/2, where G(k) = 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013
G.f.: exp( Sum_{n>=1} (x^n/n) / (1 + (-x)^n) ). - Paul D. Hanna, Jul 24 2013
a(n) ~ Pi * BesselI(1, sqrt(8*n + 1)*Pi/4) / (2*sqrt(8*n + 1)) ~ exp(Pi*sqrt(n/2)) / (4 * (2*n)^(3/4)) * (1 + (Pi/(16*sqrt(2)) - 3/(4*Pi*sqrt(2))) / sqrt(n) + (Pi^2/1024 - 15/(64*Pi^2) - 15/128) / n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A046897(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082303. - Michael Somos, Sep 30 2017

More terms from James Sellers

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

A104502 Number of partitions where no part is a multiple of 9.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 74, 96, 128, 165, 216, 275, 354, 447, 569, 712, 896, 1113, 1388, 1712, 2117, 2595, 3186, 3882, 4735, 5739, 6959, 8392, 10121, 12150, 14582, 17429, 20823, 24789, 29494, 34979, 41456, 48993, 57856, 68148, 80204

Offset: 0

Eric W. Weisstein, Mar 11 2005

nonn

Coefficients of the B-Dyson Mod 27 identity.

Also partitions where parts are repeated at most 8 times. - Joerg Arndt, Dec 31 2012

			G.f. = 1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 11*q^6 + 15*q^7 + 22*q^8 + 29*q^9 + ...
B(q) = q + q^4 + 2*q^7 + 3*q^10 + 5*q^13 + 7*q^16 + 11*q^19 + 15*q^22 + ...

F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988, see p. 15, eq. (11).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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, Dyson Mod 27 Identities

Cf. A062246, A104501, A104503, A104504.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
Cf. A112193, A261733, A320611.

seq(coeff(series(mul((1-x^(9*k))/(1-x^k),k=1..n),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Sep 29 2018

nmax = 50; CoefficientList[Series[Product[(1 - x^(9*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
a[n_] := a[n] = (1/n) Sum[DivisorSum[k, Boole[!Divisible[#, 9]] #&] a[n-k], {k, 1, n}]; a[0] = 1;
a /@ Range[0, 50] (* Jean-François Alcover, Oct 01 2019, after Seiichi Manyama *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 9], 0, 2] ], {n, 0, 46}] (* Robert Price, Jul 29 2020 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^9 + A) / eta(x + A), n))}; /* Michael Somos, Jan 09 2006 */

{A116607(n)=sigma(n)-if(n%9==0, 9*sigma(n/9))}
{a(n)=polcoeff(exp(sum(k=1, n+1, A116607(k)*x^k/k+x*O(x^n))), n)} /* Paul D. Hanna, Jun 13 2011 */

Expansion of q^(-1/3) * eta(q^9) / eta(q) in powers of q. - Michael Somos, Jan 09 2006
Euler transform of period 9 sequence [1, 1, 1, 1, 1, 1, 1, 1, 0, ...]. - Michael Somos, Jan 09 2006
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^3 + v^3 - u*v - 3*(u*v)^2. - Michael Somos, Jan 09 2006
G.f.: Product_{k>0} (1-x^(9k))/(1-x^k) = 1 + 1/(1-x)*(Sum_{k>0} x^(k^2+k) Product_{i=1..k} (1+x^i+x^(2i))/((1-x^(2i))*(1-x^(2i+1))))
G.f. A(x) = 1/g.f. A062246.
Logarithmic derivative yields A116607 (sum of the divisors of n which are not divisible by 9). - Paul D. Hanna, Jun 13 2011
a(n) ~ 2*Pi * BesselI(1, 4*sqrt(3*n + 1) * Pi/9) / (9*sqrt(3*n + 1)) ~ exp(4*Pi*sqrt(n/3)/3) / (sqrt(2) * 3^(7/4) * n^(3/4)) * (1 + (2*Pi/(9*sqrt(3)) - 9*sqrt(3)/(32*Pi)) / sqrt(n) + (2*Pi^2/243 - 405/(2048*Pi^2) - 5/16) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A116607(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f. is a period 1 Fourier series that satisfies f(-1 / (81 t)) = 1/3 g(t) where g() is the g.f. for A062246. - Michael Somos, Jun 27 2017

Simplified definition. - N. J. A. Sloane, Oct 20 2019

A219601 Number of partitions of n in which no parts are multiples of 6.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 65, 85, 111, 143, 184, 234, 297, 374, 470, 586, 729, 902, 1113, 1367, 1674, 2042, 2485, 3013, 3645, 4395, 5288, 6344, 7595, 9070, 10809, 12852, 15252, 18062, 21352, 25191, 29671, 34884, 40948, 47985, 56146, 65592

Offset: 0

Arkadiusz Wesolowski, Nov 23 2012

Also partitions where parts are repeated at most 5 times. [Joerg Arndt, Dec 31 2012]

			7 = 7
  = 5 + 2
  = 5 + 1 + 1
  = 4 + 3
  = 4 + 2 + 1
  = 4 + 1 + 1 + 1
  = 3 + 3 + 1
  = 3 + 2 + 2
  = 3 + 2 + 1 + 1
  = 3 + 1 + 1 + 1 + 1
  = 2 + 2 + 2 + 1
  = 2 + 2 + 1 + 1 + 1
  = 2 + 1 + 1 + 1 + 1 + 1
  = 1 + 1 + 1 + 1 + 1 + 1 + 1
so a(7) = 14.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Arkadiusz Wesolowski)
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

Cf. A097797.
Cf. A261770, A261736, A320608.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

m = 47; f[x_] := (x^6 - 1)/(x - 1); g[x_] := Product[f[x^k], {k, 1, m}]; CoefficientList[Series[g[x], {x, 0, m}], x] (* Arkadiusz Wesolowski, Nov 27 2012 *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 6], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)

for(n=0, 47, A=x*O(x^n); print1(polcoeff(eta(x^6+A)/eta(x+A), n), ", "))

G.f.: P(x^6)/P(x), where P(x) = prod(k>=1, 1-x^k).
a(n) ~ Pi*sqrt(5) * BesselI(1, sqrt(5*(24*n + 5)/6) * Pi/6) / (3*sqrt(24*n + 5)) ~ exp(Pi*sqrt(5*n)/3) * 5^(1/4) / (12 * n^(3/4)) * (1 + (5^(3/2)*Pi/144 - 9/(8*Pi*sqrt(5))) / sqrt(n) + (125*Pi^2/41472 - 27/(128*Pi^2) - 25/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A284326(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A035985 Number of partitions of n into parts not a multiple of 7. Also number of partitions with at most 6 parts of size 1 and differences between parts at distance 9 are greater than 1.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 70, 90, 119, 153, 199, 252, 324, 406, 515, 642, 804, 994, 1236, 1517, 1869, 2282, 2791, 3387, 4118, 4970, 6006, 7217, 8673, 10374, 12411, 14780, 17601, 20883, 24766, 29274, 34588, 40741, 47964, 56319, 66080, 77350

Offset: 0

Olivier Gérard

Case k=10, i=7 of Gordon Theorem.

			B(x) = x +x^5 +2*x^9 +3*x^13 +5*x^17 +7*x^21 +11*x^25 +14*x^29 +...

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000
GDZ, Digitized volumes of Crelle
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 über Zerfällungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. This sequence arises as the coefficients of Y = C/B on p. 118.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Cf. A320609.

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^7]/QP[q] + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 7], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^7+A)/eta(x+A), n))} /* Michael Somos, Jan 17 2006 */

Vec(prod(k=1, 50, (1 - x^(7*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017

A035985_upto(N,q='x+O('x^N))=Vec(eta(q^7)/eta(q)) \\ M. F. Hasler, Dec 09 2019

Euler transform of period 7 sequence [1, 1, 1, 1, 1, 1, 0, ...]. - Michael Somos, Jan 17 2006
Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(u^4+v^4)-u*v*(1+3*u*v+7*(u*v)^2).
G.f.: Product_{k>0} (1-x^(7k))/(1-x^k).
Given g.f. A(x) then B(x)=x*A(x)^4 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u,v,w)= (u^2+u*w+w^2) -v -8*v*(u+v+w) -49*v^2*(u+w). - Michael Somos, May 28 2006
G.f. is product k>0 P7(x^k) where P7 is 7th cyclotomic polynomial.
Expansion of q^(-1/4)eta(q^7)/eta(q) in powers of q. - Michael Somos, Jan 17 2006
a(n) ~ 2*Pi * BesselI(1, sqrt((4*n + 1)/7) * Pi) / (7*sqrt(4*n + 1)) ~ exp(2*Pi*sqrt(n/7)) / (2 * 7^(3/4) * n^(3/4)) * (1 + (Pi/(4*sqrt(7)) - 3*sqrt(7)/(16*Pi)) / sqrt(n) + (Pi^2/224 - 105/(512*Pi^2) - 15/64) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A113957(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

Definition simplified by N. J. A. Sloane, Oct 20 2019

A261776 Expansion of Product_{k>=1} (1 - x^(10*k))/(1 - x^k).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 98, 130, 169, 220, 282, 363, 460, 584, 735, 923, 1151, 1435, 1775, 2194, 2698, 3311, 4045, 4935, 5994, 7270, 8787, 10600, 12749, 15310, 18330, 21912, 26130, 31107, 36949, 43823, 51863, 61290, 72293, 85145, 100107

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

General asymptotic formula (Hagis, 1971): If s > 1 and g.f. = Product_{k>=1} (1 - x^(s*k))/(1 - x^k), then a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), minor asymptotic terms added by Vaclav Kotesovec, Jan 13 2017
The formula in the article by Noureddine Chair: "The Euler-Riemann Gases, and Partition Identities", p. 32, is incorrect (must be s -> s-1 and 24 -> 24*n).
Number of partitions in which no part occurs more than 9 times. - Ilya Gutkovskiy, May 31 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Noureddine Chair, The Euler-Riemann Gases, and Partition Identities, arXiv:1306.5415 [math-ph], 2013, p. 32.
Peter Hagis jr., Partitions with a restriction on the multiplicity of the summands, Transactions of the American Mathematical Society, Volume 155, Number 2, April 1971.
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.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Cf. A261772, A145707, A320612.

nmax = 50; CoefficientList[Series[Product[(1 - x^(10*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 10], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 29 2020 *)

Vec(prod(k=1, 51, (1 - x^(10*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017

a(n) ~ 3*Pi * BesselI(1, sqrt((24*n + 9)/10) * Pi/2) / (5*sqrt(24*n + 9)) ~ exp(Pi*sqrt(3*n/5)) * 3^(1/4) / (4 * 5^(3/4) * n^(3/4)) * (1 + (3^(3/2)*Pi/(16*sqrt(5)) - sqrt(15)/(8*Pi)) / sqrt(n) + (27*Pi^2/2560 - 25/(128*Pi^2) - 45/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017

a(n) = (1/n)*Sum_{k=1..n} A284344(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A261775 Expansion of Product_{k>=1} (1 - x^(8*k))/(1 - x^k).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72, 94, 124, 161, 208, 266, 341, 431, 545, 684, 856, 1064, 1322, 1631, 2009, 2464, 3014, 3672, 4467, 5411, 6543, 7888, 9489, 11383, 13632, 16280, 19409, 23088, 27415, 32483, 38430, 45371, 53485, 62939, 73950, 86742

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

Number of partitions in which no part occurs more than 7 times. - Ilya Gutkovskiy, May 31 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 30
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.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Cf. A261771, A261735, A320610.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
     signum(irem(d, 8)), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 07 2022

nmax = 50; CoefficientList[Series[Product[(1 - x^(8*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 8], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)

Vec(prod(k=1, 51, (1 - x^(8*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017

a(n) ~ Pi*sqrt(7) * BesselI(1, sqrt(7*(24*n + 7)/8) * Pi/6) / (4*sqrt(24*n + 7)) ~ exp(Pi*sqrt(7*n/3)/2) * 7^(1/4) / (2^(7/2) * 3^(1/4) * n^(3/4)) * (1 + (7^(3/2)*Pi/(96*sqrt(3)) - 3*sqrt(3)/(4*Pi*sqrt(7))) / sqrt(n) + (343*Pi^2/55296 - 45/(224*Pi^2) - 35/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A284341(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f.: A(x)*A(x^2)*A(x^4) where A(x) is the o.g.f. for A000009. (see Flajolet, Sedgewick link) - Geoffrey Critzer, Aug 07 2022

A328546 Number of 12-regular partitions of n (no part is a multiple of 12).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 100, 133, 173, 226, 290, 374, 475, 605, 762, 960, 1199, 1497, 1856, 2299, 2831, 3482, 4261, 5208, 6337, 7700, 9321, 11266, 13572, 16325, 19578, 23444, 27999, 33389, 39721, 47185, 55929, 66199, 78199, 92246

Offset: 0

N. J. A. Sloane, Oct 19 2019

nonn

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

f:=(k,M) -> mul(1-q^(k*j),j=1..M);
LRP := (L,M) -> f(L,M)/f(1,M);
s := L -> seriestolist(series(LRP(L,80),q,60));
s(12);

Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 12], 0, 2] ], {n, 0, 46}] (* Robert Price, Jul 28 2020 *)

a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), set s=12. - Vaclav Kotesovec, Aug 01 2022

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

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A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

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

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A104502 Number of partitions where no part is a multiple of 9.

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