A261771 - OEIS

%I #15 Feb 16 2018 10:09:29

%S 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,

%T 118,136,156,177,203,231,263,299,338,383,433,489,550,620,696,781,877,

%U 981,1097,1227,1369,1526,1702,1893,2104,2339,2595,2877,3189,3530

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

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

%H Alois P. Heinz, <a href="/A261771/b261771.txt">Table of n, a(n) for n = 0..10000</a>

%F 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

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

%p with(numtheory):

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

%p        [0, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1]

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

%p     end:

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Aug 31 2015

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

%Y Cf. A261735.

%Y 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).

%Y Column k=8 of A290307.

%K nonn

%O 0,4

%A _Vaclav Kotesovec_, Aug 31 2015