A341241 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^3.

1, 0, 3, 3, 6, 9, 13, 21, 27, 40, 54, 75, 97, 129, 171, 220, 282, 360, 460, 576, 720, 896, 1116, 1374, 1682, 2061, 2517, 3050, 3684, 4449, 5354, 6414, 7656, 9135, 10875, 12891, 15243, 18015, 21243, 24966, 29286, 34326, 40156, 46851, 54573, 63509, 73794, 85551, 99035, 114555

Offset: 3

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

• nonn

Crossrefs

Cf. A000700, A022598, A047655, A107635, A327381, A338463, A341221, A341243, A341244, A341245, A341246, A341247, A341251.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..52);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax = 52; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

Formula

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^3.

a(n) ~ A107635(n). - Vaclav Kotesovec, Feb 20 2021