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

1,7

Links

Table of n, a(n) for n=1..55.

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)).

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

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

Crossrefs

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

Sequence in context: A077141 A276566 A158624 * A021659 A011506 A054400

Adjacent sequences: A363235 A363236 A363237 * A363239 A363240 A363241

Keyword

nonn

Author

Seiichi Manyama, May 23 2023

Status

approved