A374977 a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l).

0, 0, 0, 1, 12, 70, 280, 885, 2364, 5586, 12000, 23870, 44660, 79272, 134768, 220565, 349440, 538270, 807840, 1187004, 1706840, 2415150, 3354120, 4601870, 6209612, 8303610, 10935960, 14309640, 18460260, 23708184, 30044000, 37967925, 47368480, 59022432, 72633816

Offset: 1

Author

Chai Wah Wu, Jul 26 2024

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000385, A319083, A374951.

Programs

Mathematica

b[n_, k_] := b[n, k] = If[k == 0, If[n == 0, 1, 0], If[k == 1, If[n == 0, 0, DivisorSigma[1, n]], Function[q, Sum[b[j, q]*b[n - j, k - q], {j, 0, n}]][Quotient[k, 2]]]];
a[n_] := b[n, 4];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Jul 11 2025, after Alois P. Heinz in A319083 *)

Python

from sympy import divisor_sigma
def A374977(n): return sum((5*divisor_sigma(i+1,3)-(5+6*i)*divisor_sigma(i+1))*(5*divisor_sigma(n-i-1,3)-(5+6*(n-i-2))*divisor_sigma(n-i-1)) for i in range(1,n-2))//144

Formula

4-fold convolution of A000203.

Convolution of A000203 and A374951.

Convolution of A000385 with itself.

a(n) = Sum_{i=1..n-1} A000203(i)*A374951(n-i).

a(n) = Sum_{i=1..n-3} A000385(i)*A000385(n-i-2).

Column k=4 of A319083.

Sum_{k=1..n} a(k) ~ Pi^8 * n^8 / 52254720. - Vaclav Kotesovec, Sep 20 2024