A363233 Number of partitions of n with rank a multiple of 4.

1, 0, 1, 1, 3, 1, 5, 4, 10, 8, 16, 17, 29, 29, 48, 53, 81, 89, 130, 149, 208, 238, 325, 381, 506, 592, 770, 910, 1165, 1374, 1738, 2057, 2571, 3038, 3761, 4451, 5461, 6447, 7855, 9270, 11219, 13214, 15899, 18703, 22386, 26276, 31306, 36691, 43525, 50902, 60149, 70221, 82679, 96325

Offset

1,5

Links

Table of n, a(n) for n=1..54.

Formula

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^(4*k)) / (1-x^(4*k)).

Maple

b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 4)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..54);  # Alois P. Heinz, May 23 2023

Prog

(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^(4*k))/(1-x^(4*k))))

Crossrefs

Cf. A000041, A328988, A340601, A363237, A363238, A363239.

Sequence in context: A010847 A331698 A082129 * A159970 A096374 A007085

Adjacent sequences: A363230 A363231 A363232 * A363234 A363235 A363236

Keyword

nonn

Author

Seiichi Manyama, May 23 2023

Status

approved