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A261770
Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(6*k)).
9
1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 11, 13, 16, 19, 22, 26, 30, 35, 41, 47, 55, 63, 73, 84, 96, 110, 125, 143, 162, 184, 208, 235, 266, 300, 338, 380, 427, 479, 536, 600, 670, 748, 834, 929, 1034, 1149, 1277, 1417, 1571, 1740, 1925, 2129, 2351, 2596, 2863
OFFSET
0,4
COMMENTS
a(n) is the number of partitions of n into distinct parts where no part is a multiple of 6. - Joerg Arndt, Aug 31 2015
LINKS
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. 12.
FORMULA
a(n) ~ exp(Pi*sqrt(5*n/2)/3) * 5^(1/4) / (2^(7/4) * sqrt(3) * n^(3/4)) * (1 - (9/(4*Pi*sqrt(10)) + 5*Pi*sqrt(5/2)/144) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017
G.f.: Product_{k>=1} (1 - x^(12*k-6))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017
MAPLE
b:= proc(n, i) option remember; local r;
`if`(2*n>i*(i+1)-(j-> 6*j*(j+1))(iquo(i, 6, 'r')), 0,
`if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..80); # Alois P. Heinz, Aug 31 2015
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(6*k)), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
Cf. A261736.
Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A097793 (m=7), A261771 (m=8), A112193 (m=9), A261772 (m=10).
Column k=6 of A290307.
Sequence in context: A305148 A058724 A029021 * A096792 A335746 A015741
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Aug 31 2015
STATUS
approved