login
Number of partitions of n with rank a multiple of 6.
3

%I #10 May 23 2023 10:46:48

%S 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 340,392,514,610,771,916,1166,1367,1711,2032,2503,2965,3647,4293,5237,

%U 6188,7469,8808,10613,12459,14920,17530,20862,24457,29029,33924,40099,46829,55101,64215,75386

%N Number of partitions of n with rank a multiple of 6.

%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^(6*k)) / (1-x^(6*k)).

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

%p `if`(irem(i-c, 6)=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^(6*k))/(1-x^(6*k))))

%Y Cf. A000041, A328988, A340601, A363233, A363237, A363239.

%K nonn

%O 1,7

%A _Seiichi Manyama_, May 23 2023