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A363238 Number of partitions of n with rank a multiple of 6.

%I A363238 #10 May 23 2023 10:46:48
%S A363238 1,0,1,1,1,1,5,2,6,6,10,11,21,19,32,37,51,59,90,97,138,162,215,253,
%T A363238 340,392,514,610,771,916,1166,1367,1711,2032,2503,2965,3647,4293,5237,
%U A363238 6188,7469,8808,10613,12459,14920,17530,20862,24457,29029,33924,40099,46829,55101,64215,75386
%N A363238 Number of partitions of n with rank a multiple of 6.
%F A363238 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^(6*k)) / (1-x^(6*k)).
%p A363238 b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
%p A363238      `if`(irem(i-c, 6)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
%p A363238     end:
%p A363238 a:= n-> b(n, 1$2):
%p A363238 seq(a(n), n=1..55);  # _Alois P. Heinz_, May 23 2023
%o A363238 (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^(6*k))/(1-x^(6*k))))
%Y A363238 Cf. A000041, A328988, A340601, A363233, A363237, A363239.
%K A363238 nonn
%O A363238 1,7
%A A363238 _Seiichi Manyama_, May 23 2023