login
A261771
Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(8*k)).
9
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
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
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.
Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A112193 (m=9), A261772 (m=10).
Column k=8 of A290307.
Sequence in context: A367214 A342499 A325096 * A015743 A015755 A096443
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Aug 31 2015
STATUS
approved