A319647 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^sigma_n(k).

1, 1, 6, 38, 526, 13074, 702813, 70939556, 13879861574, 5583837482767, 4393101918607162, 6717450870069292051, 21057681806321501744772, 131246096280071506595491449, 1604095619160115980216291007253, 40299198842857238408636666363954678, 2031474817845087309816967328335309651478

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..80

Formula

a(n) = [x^n] Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(j^n).

a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^k/(k*(1 - x^k))).

Maple

with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      sigma[k](d), d=divisors(j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 26 2018

Mathematica

Table[SeriesCoefficient[Product[1/(1 - x^k)^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[Product[1/(1 - x^(i j))^(j^n), {j, 1, n}], {i, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

Prog

(PARI) {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^n))^sigma(k, n)), n)} \\ Seiichi Manyama, Oct 27 2018

Crossrefs

Cf. A006171, A061256, A275585, A288391, A301542, A301543, A301544, A301545, A301546, A301547, A321042.

Sequence in context: A367838 A319194 A303865 * A239983 A375346 A096674

Adjacent sequences: A319644 A319645 A319646 * A319648 A319649 A319650

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 26 2018

Status

approved