A277090 Expansion of Product_{k>=0} 1/(1 - x^(8*k+1)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 12, 12, 13, 15, 17, 18, 19, 19, 19, 19, 20, 23, 26, 28, 29, 30, 30, 30, 31, 34, 38, 41, 43, 44, 45, 45, 46, 50, 55, 60, 63, 65, 66, 67, 68, 72, 79, 85, 90, 93, 95, 96, 98, 103, 111, 120, 127, 132, 135, 137, 139, 145

Offset: 0

Author

Ilya Gutkovskiy, Sep 29 2016

Keywords

• nonn

Comments

Number of partitions of n into parts congruent to 1 mod 8.

More generally, the ordinary generating function for the number of partitions of n into parts congruent to 1 mod m (for m>0) is Product_{k>=0} 1/(1 - x^(m*k+1)).

Examples

			a(10) = 2, because we have [9, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• 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.
• Index entries for related partition-counting sequences

Crossrefs

Cf. A017077, A284100.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), A035451 (m=4), A109697 (m=5), A109701 (m=6), A109703 (m=7).

Programs

Mathematica

CoefficientList[Series[QPochhammer[x, x^8]^(-1), {x, 0, 90}], x]

Formula

G.f.: Product_{k>=0} 1/(1 - x^(8*k+1)).

a(n) ~ exp((Pi*sqrt(n))/(2*sqrt(3)))*Gamma(1/8)/(4*3^(1/16)*(2*Pi)^(7/8)*n^(9/16)).

a(n) = (1/n)*Sum_{k=1..n} A284100(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017