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

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

Author

Vaclav Kotesovec, Aug 31 2015

Keywords

• nonn

Comments

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

Links

• 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.

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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