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A277090 Expansion of Product_{k>=0} 1/(1 - x^(8*k+1)). 6

%I

%S 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,

%T 10,11,12,12,12,12,12,13,15,17,18,19,19,19,19,20,23,26,28,29,30,30,30,

%U 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

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

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

%C 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)).

%H Seiichi Manyama, <a href="/A277090/b277090.txt">Table of n, a(n) for n = 0..10000</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015.

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

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

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

%F a(n) = (1/n)*Sum_{k=1..n} A284100(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 20 2017

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

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

%Y Cf. A017077, A284100.

%Y 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).

%K nonn

%O 0,10

%A _Ilya Gutkovskiy_, Sep 29 2016

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Last modified February 18 13:07 EST 2018. Contains 299322 sequences. (Running on oeis4.)