A340992 a(n) is the (2n)-th term of the n-fold self-convolution of the number of divisors function tau.

1, 2, 8, 41, 216, 1172, 6491, 36430, 206472, 1179104, 6774048, 39107400, 226683903, 1318427762, 7690414740, 44970645116, 263545466456, 1547445069318, 9101515979306, 53613206171619, 316243949777696, 1867702439169958, 11042787840419398, 65357054283015120

Offset: 0

Author

Alois P. Heinz, Feb 01 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1281

Crossrefs

Cf. A000005, A320019.

Programs

Maple

b:= proc(n, k) option remember; `if`(k=0, 1,
      `if`(k=1, numtheory[tau](n+1), (q->
       add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);

Mathematica

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0], If[k == 1, If[n == 0, 0, DivisorSigma[0, n]], With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n - j, k - q], {j, 0, n}]]]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz in A320019 *)

Formula

a(n) = [x^(2n)] (Sum_{j>=1} tau(j)*x^j)^n.

a(n) = A320019(2n,n).