A261771 - OEIS

A261771

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

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 13, 15, 18, 22, 26, 30, 36, 42, 49, 58, 67, 77, 89, 103, 118, 136, 156, 177, 203, 231, 263, 299, 338, 383, 433, 489, 550, 620, 696, 781, 877, 981, 1097, 1227, 1369, 1526, 1702, 1893, 2104, 2339, 2595, 2877, 3189, 3530

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OFFSET

0,4

COMMENTS

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 8.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

FORMULA

a(n) ~ exp(Pi*sqrt(7*n/6)/2) * 7^(1/4) / (4 * 6^(1/4) * n^(3/4)) * (1 - (3*sqrt(3)/ (2*Pi*sqrt(14)) + 7*Pi*sqrt(7)/(96*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

G.f.: Product_{k>=1} (1 - x^(16*k-8))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017

MAPLE

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

[0, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1]

[1+irem(d, 16)], d=divisors(j))*a(n-j), j=1..n)/n)

end:

seq(a(n), n=0..80); # Alois P. Heinz, Aug 31 2015

MATHEMATICA

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(8*k)), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A261735.