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A363047 Number of partitions of n whose greatest part is a multiple of 5.

%I A363047 #18 May 21 2023 10:51:40
%S A363047 1,0,0,0,0,1,1,2,3,5,8,11,15,21,28,38,49,64,82,105,134,168,211,263,
%T A363047 327,406,501,616,757,926,1133,1378,1676,2031,2460,2970,3581,4306,5173,
%U A363047 6197,7419,8855,10561,12565,14934,17712,20982,24805,29294,34529,40658,47785,56104
%N A363047 Number of partitions of n whose greatest part is a multiple of 5.
%H A363047 Seiichi Manyama, <a href="/A363047/b363047.txt">Table of n, a(n) for n = 0..1000</a>
%F A363047 G.f.: Sum_{k>=0} x^(5*k)/Product_{j=1..5*k} (1-x^j).
%F A363047 a(n) ~ A000041(n)/5. - _Vaclav Kotesovec_, May 21 2023
%p A363047 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A363047      `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
%p A363047     end:
%p A363047 a:= n-> add(b(n-5*i, min(n-5*i, 5*i)), i=0..n/5):
%p A363047 seq(a(n), n=0..52);  # _Alois P. Heinz_, May 14 2023
%o A363047 (PARI) my(N=60, x='x+O('x^N)); Vec(sum(k=0, N, x^(5*k)/prod(j=1, 5*k, 1-x^j)))
%Y A363047 Column 5 of A363048.
%K A363047 nonn
%O A363047 0,8
%A A363047 _Seiichi Manyama_, May 14 2023