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

1, 0, 1, 1, 1, 1, 5, 2, 6, 6, 10, 11, 21, 19, 32, 37, 51, 59, 90, 97, 138, 162, 215, 253, 340, 392, 514, 610, 771, 916, 1166, 1367, 1711, 2032, 2503, 2965, 3647, 4293, 5237, 6188, 7469, 8808, 10613, 12459, 14920, 17530, 20862, 24457, 29029, 33924, 40099, 46829, 55101, 64215, 75386

Offset: 1

Author

Seiichi Manyama, May 23 2023

Keywords

• nonn

Crossrefs

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

Programs

Maple

b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 6)=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..55);  # Alois P. Heinz, May 23 2023

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))))

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