A363239 - OEIS

%I #10 May 23 2023 10:46:43

%S 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 348,430,532,653,800,978,1193,1449,1758,2127,2569,3091,3717,4455,5334,

%U 6369,7596,9039,10739,12734,15080,17822,21039,24791,29176,34277,40227,47133,55165,64468

%N Number of partitions of n with rank a multiple of 7.

%F 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 b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,

%p      `if`(irem(i-c, 7)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))

%p     end:

%p a:= n-> b(n, 1$2):

%p seq(a(n), n=1..55);  # _Alois P. Heinz_, May 23 2023

%o (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 Cf. A000041, A328988, A340601, A363233, A363237, A363238.

%K nonn

%O 1,7

%A _Seiichi Manyama_, May 23 2023