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A363237 Number of partitions of n with rank a multiple of 5.

%I A363237 #10 May 23 2023 10:46:58
%S A363237 1,0,1,1,1,3,3,4,6,8,12,15,21,27,34,47,59,77,98,125,160,200,251,315,
%T A363237 390,488,602,744,913,1120,1370,1669,2029,2462,2975,3597,4327,5203,
%U A363237 6237,7466,8919,10634,12653,15035,17824,21114,24950,29455,34705,40844,47991,56317,65987,77231,90252
%N A363237 Number of partitions of n with rank a multiple of 5.
%F A363237 G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(5*k)) / (1-x^(5*k)).
%p A363237 b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
%p A363237      `if`(irem(i-c, 5)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
%p A363237     end:
%p A363237 a:= n-> b(n, 1$2):
%p A363237 seq(a(n), n=1..55);  # _Alois P. Heinz_, May 23 2023
%o A363237 (PARI) my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(5*k))/(1-x^(5*k))))
%Y A363237 Cf. A000041, A328988, A340601, A363233, A363238, A363239.
%K A363237 nonn
%O A363237 1,6
%A A363237 _Seiichi Manyama_, May 23 2023