A320019 -id:A320019 - cp's OEIS frontend

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

Alois P. Heinz, Feb 01 2021

nonn

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

Cf. A000005, A320019.

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);

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 *)

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

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

A375002 a(n) = Sum_{i+j+k+m=n, i,j,k,m >= 1} tau(i) * tau(j) * tau(k) * tau(m).

Original entry on oeis.org

0, 0, 0, 1, 8, 32, 92, 216, 440, 814, 1392, 2244, 3452, 5096, 7292, 10129, 13760, 18284, 23868, 30662, 38820, 48556, 59948, 73424, 88796, 106886, 127052, 150732, 176560, 206920, 239344, 277616, 317516, 365034, 413508, 471637, 529712, 600076, 668708, 753070, 833408

Offset: 1

Seiichi Manyama, Jul 27 2024

nonn

4-fold convolution of tau (A000005).

Column k=4 of A320019.

Cf. A000005, A055507.

my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^k/(1-x^k))^4))

G.f.: ( Sum_{k>=1} x^k/(1 - x^k) )^4.

a(n) = Sum_{i=1..n-3} A055507(i)*A055507(n-2-i). - Chai Wah Wu, Jul 27 2024

cp's OEIS Frontend

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

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