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A363239 Number of partitions of n with rank a multiple of 7.

%I A363239 #10 May 23 2023 10:46:43
%S A363239 1,0,1,1,1,1,3,4,4,6,8,11,15,19,26,33,43,55,70,89,114,144,179,225,280,
%T A363239 348,430,532,653,800,978,1193,1449,1758,2127,2569,3091,3717,4455,5334,
%U A363239 6369,7596,9039,10739,12734,15080,17822,21039,24791,29176,34277,40227,47133,55165,64468
%N A363239 Number of partitions of n with rank a multiple of 7.
%F A363239 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^(7*k)) / (1-x^(7*k)).
%p A363239 b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
%p A363239      `if`(irem(i-c, 7)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
%p A363239     end:
%p A363239 a:= n-> b(n, 1$2):
%p A363239 seq(a(n), n=1..55);  # _Alois P. Heinz_, May 23 2023
%o A363239 (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^(7*k))/(1-x^(7*k))))
%Y A363239 Cf. A000041, A328988, A340601, A363233, A363237, A363238.
%K A363239 nonn
%O A363239 1,7
%A A363239 _Seiichi Manyama_, May 23 2023