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

2,2

Links

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

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

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 *)

Crossrefs

Column k=2 of A257673.

Cf. A000219, A161870.

Sequence in context: A256569 A132130 A022571 * A291226 A228704 A320855

Adjacent sequences: A321944 A321945 A321946 * A321948 A321949 A321950

Keyword

nonn

Author

Alois P. Heinz, Nov 22 2018

Status

approved