A321947 Column k=2 of triangle A257673.

1, 6, 21, 62, 162, 396, 917, 2036, 4380, 9152, 18694, 37380, 73444, 141918, 270370, 508178, 943876, 1733468, 3151396, 5674152, 10126435, 17921016, 31468623, 54848750, 94935565, 163232096, 278903915, 473693432, 799949111, 1343550666, 2244807927, 3731885232

Offset: 2

Author

Alois P. Heinz, Nov 22 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..10000

Crossrefs

Column k=2 of A257673.

Cf. A000219, A161870.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
seq(a(n), n=2..35);

Mathematica

A321947[n_] := Module[{nn = n}, SeriesCoefficient[Product[1/(1 - x^i)^(2 i), {i, 1, nn}], {x, 0, nn}] - 2*SeriesCoefficient[Product[(1 - x^k)^-k, {k, nn}], {x, 0, nn}]]; Table[A321947[n], {n, 2, 33}] (* Robert P. P. McKone, Jan 30 2021 *)
b[n_, k_] := b[n, k] = If[n == 0, 1, k*Sum[
     b[n - j, k]*DivisorSigma[2, j], {j, 1, n}]/n];
a[n_] := With[{k = 2}, Sum[b[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]];
Table[a[n], {n, 2, 35}] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

Formula

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^2. - Ilya Gutkovskiy, Jan 30 2021

a(n) = A161870(n) - 2*A000219(n). - Vaclav Kotesovec, Jan 30 2021