A035386 -id:A035386 - cp's OEIS frontend

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

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

A035382 Number of partitions of n into parts congruent to 1 mod 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 26, 29, 33, 38, 42, 48, 54, 61, 68, 77, 85, 96, 107, 119, 132, 148, 163, 181, 201, 223, 245, 272, 299, 330, 363, 400, 438, 483, 529, 580, 635, 697, 760, 832, 908, 992, 1081, 1180, 1283, 1399, 1521

Offset: 0

Olivier Gérard

nonn

a(n) = A116373(3*n). - Reinhard Zumkeller, Feb 15 2006

			a(3) = 1 because we have [1,1,1];
a(4) = 2 because we have [1,1,1,1] and [4];
a(9) = 4 because we have [7,1,1], [4,4,1], [4,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Joerg Arndt, Matters Computational (The Fxtbook), p. 350.

Cf. A035386, A035451, A261612.

g:= 1/product(1-x^(1+3*j), j=0..50): gser:= series(g, x=0, 64): seq(coeff(gser, x, n), n=0..61); # Emeric Deutsch, Mar 30 2006
# second Maple program
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-3) +`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(n, 3*iquo(n, 3)+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 03 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-3] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[n, 3*Quotient[n, 3]+1]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = -1; Do[If[Mod[k, 3] == 1, Do[poly[[j + 1]] -= poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly2 = Take[poly, {2, nmax + 1}]; poly3 = 1 + Sum[poly2[[n]]*x^n, {n, 1, Length[poly2]}]; CoefficientList[Series[1/poly3, {x, 0, Length[poly2]}], x] (* Vaclav Kotesovec, Jan 13 2017 *)
nmax = 50; s = Range[0, nmax/3]*3 + 1;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

a(n) = 1/n*Sum_{k=1..n} A078181(k)*a(n-k), a(0) = 1.
G.f.: 1/prod(j>=0, 1-x^(1+3*j) ). - Emeric Deutsch, Mar 30 2006
From Joerg Arndt, Oct 02 2012: (Start)
G.f.: sum(n>=0, q^n/prod(k=1..n, 1-q^(3*k)) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(R*n)/prod(k=1..n, 1-q^(M*k) ) ) for partitions into parts R mod M (where R!=0).
G.f. sum(n>=0, q^(3*n^2-2*n) / prod(k=0..n-1, (1-q^(3*k+3))*(1-q^(3*k+1))) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(M*n^2+(R-M)*n) / prod(k=0..n-1, (1-q^(M*k+M))*(1-q^(M*k+R))) ) for partitions into parts R mod M (where R!=0). (See Fxtbook link)
(End)
a(n) ~ Gamma(1/3) * exp(sqrt(2*n)*Pi/3) / (2*sqrt(3) * (2*Pi*n)^(2/3)) * (1 + (Pi/72 - 2/(3*Pi)) / sqrt(2*n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
Euler transform of period 3 sequence [ 1, 0, 0, ...]. - Kevin T. Acres, Apr 28 2018

A078182 a(n) = Sum_{d|n, d == 2 (mod 3)} d.

Original entry on oeis.org

0, 2, 0, 2, 5, 2, 0, 10, 0, 7, 11, 2, 0, 16, 5, 10, 17, 2, 0, 27, 0, 13, 23, 10, 5, 28, 0, 16, 29, 7, 0, 42, 11, 19, 40, 2, 0, 40, 0, 35, 41, 16, 0, 57, 5, 25, 47, 10, 0, 57, 17, 28, 53, 2, 16, 80, 0, 31, 59, 27, 0, 64, 0, 42, 70, 13, 0, 87, 23, 56, 71, 10, 0, 76, 5, 40, 88, 28, 0, 115

Offset: 1

Vladeta Jovovic, Nov 21 2002

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001817, A001822, A035386, A078181, A272715, A353908.

A078182 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(d,3) =2 then
            a :=a+d ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, May 11 2016

a[n_] := Plus @@ Select[Divisors[n], Mod[#, 3] == 2 &]; Array[a, 100] (* Giovanni Resta, May 11 2016 *)

a(n) = sumdiv(n, d, d*((d%3) == 2)); \\ Michel Marcus, May 11 2016

G.f.: Sum_{n>=0} (3*n+2)*x^(3*n+2)/(1-x^(3*n+2)).
A078181(n) + a(n) + 3*A000203(n/3) = A000203(n), where A000203 is defined as zero for non-integer arguments. - R. J. Mathar, May 11 2016
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/36 = 0.274155... (A353908). - Amiram Eldar, Nov 26 2023

A262928 Expansion of Product_{k>=0} (1 + x^(3*k+2)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 1, 1, 2, 1, 1, 3, 1, 2, 3, 1, 3, 4, 2, 4, 4, 2, 5, 5, 3, 7, 5, 4, 8, 6, 6, 10, 7, 7, 12, 8, 10, 14, 9, 12, 16, 11, 16, 19, 13, 19, 21, 16, 24, 25, 19, 28, 28, 23, 35, 32, 28, 40, 36, 34, 48, 42, 41, 55, 47, 49, 65, 55

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003105, A035386, A261612.

nmax=100; CoefficientList[Series[Product[(1+x^(3*k-1)),{k,1,nmax}],{x,0,nmax}],x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 3] == 2, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 13 2017 *)

a(n) ~ exp(sqrt(n)*Pi/3) / (2^(5/3)*sqrt(3)*n^(3/4)) * (1 - (Pi/144 + 9/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Oct 04 2015, extended Jan 16 2017

G.f.: Sum_{k>=0} x^(k*(3*k + 1)/2) / Product_{j=1..k} (1 - x^(3*j)). - Ilya Gutkovskiy, Nov 24 2020

A116376 Number of partitions of n into parts with digital root = 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 6, 0, 0, 5, 0, 0, 7, 0, 0, 7, 0, 0, 9, 0, 0, 9, 0, 0, 12, 0, 0, 11, 0, 0, 15, 0, 0, 15, 0, 0, 18, 0, 0, 19, 0, 0, 23, 0, 0, 23, 0, 0, 29, 0, 0, 29, 0, 0, 35, 0, 0, 37

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116377(n) - A116378(n) - A114099(n).

			a(30) = #{24+6, 15+15, 6+6+6+6+6} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

Cf. A147706.

N:= 1000: # to get a(1) to a(N)
g:= mul(1/(1-x^(6+9*j)), j=0..floor((N-6)/9)):
S:= series(g, x, N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Apr 13 2015

a(n) = A035386(floor(n/3))*0^(n mod 3).

G.f.: Product_{j>=0} 1/(1 - x^(6+9*j)). - Robert Israel, Apr 13 2015

A035462 Number of partitions of n into parts 4k-1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 4, 4, 3, 4, 5, 5, 5, 6, 7, 8, 7, 8, 11, 10, 10, 13, 14, 14, 15, 17, 19, 20, 20, 24, 27, 26, 28, 33, 35, 35, 39, 44, 46, 48, 52, 58, 62, 63, 69, 78, 80, 83, 93, 100, 104, 111, 120, 130, 137, 143, 156, 169, 175, 185, 203

Offset: 0

Olivier Gérard

Also, number of partitions into parts 8k+3 or 8k+7.

Also number of partitions of n such that 2k-1 and 2k occur with the same multiplicity. Example: a(18)=3 because we have [8,7,2,1],[6,5,4,3] and [2,2,2,2,2,2,1,1,1,1,1,1]. It is easy to find a bijection between these partitions and those described in the definition. - Emeric Deutsch, Apr 05 2006

			a(18)=3 because we have [15,3],[11,7] and [3,3,3,3,3,3].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
James Mc Laughlin, Andrew V. Sills, Peter Zimmer, Rogers-Ramanujan-Slater Type Identities , arXiv:1901.00946 [math.NT]

Cf. A035441-A035468, A050452.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), this sequence (m=4), A109700 (m=5), A109702 (m=6), A109708 (m=7).

g:=1/product(1-x^(4*i-1),i=1..50): gser:=series(g,x=0,80): seq(coeff(gser,x,n),n=1..75); # Emeric Deutsch, Apr 05 2006

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+3)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)
nmax = 50; kmax = nmax/4; s = Range[0, kmax]*4 - 1;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 04 2020 *)

G.f.: 1/Product_{j>=1} (1 - x^(4*j-1)). - Emeric Deutsch, Apr 05 2006
G.f.: Sum_{n>=0} (x^(3*n) / Product_{k=1..n} (1 - x^(4*k))) = 1 + Sum_{n>=0} (x^(4*n+3) / Product_{k>=n} (1 - x^(4*k+3))) = 1 + Sum_{n>=0} (x^(4*n+3) / Product_{k=0..n} (1 - x^(4*k+3))). - Joerg Arndt, Apr 08 2011
a(n) ~ Pi^(3/4) * exp(Pi*sqrt(n/6)) / (Gamma(1/4) * 2^(13/8) * 3^(3/8) * n^(7/8)) * (1 + (Pi/(96*sqrt(6)) - 21*sqrt(3/2)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A050452(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
From Peter Bala, Feb 02 2021: (Start)
G.f.: A(x) = Sum_{n >= 0} x^(n*(4*n-1))/Product_{k = 1..n} ( (1 - x^(4*k))*(1 - x^(4*k-1)) ). (Set z = x^3 and q = x^4 in Mc Laughlin et al., Section 1.3, Entry 7.)
Similarly, A(x) =  Sum_{n >= 0} x^(n*(4*n+3))/( (1 - x^3)*Product_{k = 1..n} ((1 - x^(4*k))*(1 - x^(4*k+3))) ). (End)

Offset changed by N. J. A. Sloane, Apr 11 2010

A262946 Expansion of Product_{k>=1} 1/(1-x^(3k-1))^(3k-1).

Original entry on oeis.org

1, 0, 2, 0, 3, 5, 4, 10, 13, 15, 37, 31, 61, 87, 99, 178, 228, 286, 477, 552, 816, 1163, 1418, 2077, 2790, 3507, 5113, 6478, 8563, 11888, 15005, 20100, 27054, 34055, 46002, 59905, 76436, 102105, 130879, 168103, 221954, 281300, 363743, 472557, 597579, 772148

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

A262946(n)/A262947(n) ~ exp(3*(d1-d2)) * Gamma(1/3)^3 / (2*Pi)^(3/2), where d1 = A263030 and d2 = A263031. - Vaclav Kotesovec, Oct 08 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
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
Vaclav Kotesovec, Graph - The asymptotic ratio (120000 terms)

Cf. A035386, A262811, A262876, A262883, A262923, A262947.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=2, 3*r-1, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Oct 05 2015

nmax=60; CoefficientList[Series[Product[1/((1-x^(3k-1))^(3k-1)),{k,1,nmax}],{x,0,nmax}],x]
nmax=60; CoefficientList[Series[E^Sum[1/j*x^(2*j)*(2+x^(3*j))/(1-x^(3*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (2*Zeta(3))^(5/36) * exp(3*d1 + (3/2)^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(29/36) * Gamma(2/3) * n^(23/36)), where d1 = A263030 = Integral_{x=0..infinity} 1/x*(exp(-2*x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 1/(9*x) + exp(-x)/36) = -0.18870819197952853237641009864920797359211446726842922150941... . - Vaclav Kotesovec, Oct 08 2015

A109700 Number of partitions of n into parts each equal to 4 mod 5.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 3, 4, 2, 2, 3, 5, 4, 3, 3, 6, 6, 6, 4, 6, 7, 9, 7, 7, 8, 11, 11, 11, 9, 12, 14, 16, 13, 14, 16, 21, 20, 19, 18, 24, 26, 27, 24, 27, 31, 36, 34, 34, 35, 43, 45, 47, 43, 49, 55, 62, 58, 59, 63, 75, 77, 77, 75, 87

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(30)=2 since 30 = 14+4+4+4+4 = 9+9+4+4+4

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A284103.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), this sequence (m=5), A109702 (m=6), A109708 (m=7).

g:=1/product(1-x^(4+5*j),j=0..25): gser:=series(g,x=0,95): seq(coeff(gser,x,n),n=0..90); # Emeric Deutsch, Mar 30 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(5*k+4)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

G.f.: 1/product(1-x^(4+5j), j=0..infinity). - Emeric Deutsch, Mar 30 2006
a(n) ~ Gamma(4/5) * exp(Pi*sqrt(2*n/15)) / (2^(19/10) * 3^(2/5) * 5^(1/10) * Pi^(1/5) * n^(9/10)) * (1 - (9*sqrt(6/5)/(5*Pi) + Pi/(120*sqrt(30))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284103(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

More terms from Michael Somos, Aug 10 2005

A109702 Number of partitions of n into parts each equal to 5 mod 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 2, 1, 2, 3, 4, 4, 2, 2, 3, 5, 6, 5, 3, 3, 5, 7, 8, 6, 4, 5, 8, 10, 10, 8, 6, 8, 11, 13, 13, 10, 9, 12, 15, 18, 17, 14, 13, 16, 21, 23, 22, 18, 18, 23, 28, 31, 28, 24, 25, 31, 38, 39, 36, 32, 34

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(40)=4 since 40 = 35+5 = 29+11 = 23+17 = 5+5+5+5+5+5+5+5.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A284104.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), A109700 (m=5), this sequence (m=6), A109708 (m=7).

g:=1/product(1-x^(5+6*j),j=0..20): gser:=series(g,x=0,92): seq(coeff(gser,x,n),n=0..89); # Emeric Deutsch, Apr 14 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(6*k+5)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)
Table[Count[IntegerPartitions[n],?(Union[Mod[#,6]]=={5}&)],{n,0,90}] (* _Harvey P. Dale, Mar 08 2022 *)

G.f.: 1/product(1-x^(5+6j),j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(5/6) * exp(Pi*sqrt(n)/3) / (4 * sqrt(3) * Pi^(1/6) * n^(11/12)) * (1 - (55/(24*Pi) + Pi/144) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284104(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
Euler transform of period 6 sequence [ 0, 0, 0, 0, 1, 0, ...]. - Kevin T. Acres, Apr 28 2018

Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015

cp's OEIS Frontend

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

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A035382 Number of partitions of n into parts congruent to 1 mod 3.

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A078182 a(n) = Sum_{d|n, d == 2 (mod 3)} d.

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A262928 Expansion of Product_{k>=0} (1 + x^(3*k+2)).

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A116376 Number of partitions of n into parts with digital root = 6.

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A035462 Number of partitions of n into parts 4k-1.

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A262946 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^(3*k-1).

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A262946 Expansion of Product_{k>=1} 1/(1-x^(3k-1))^(3k-1).

A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3k-1))^(3k-1) * (1-x^(3k-2))^(3k-2)).