A374978 a(n) = Sum_{i+j+k+l+m=n, i,j,k,l,m >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l)*sigma(m).

0, 0, 0, 0, 1, 15, 110, 545, 2095, 6713, 18750, 47040, 108185, 231640, 467034, 894605, 1639680, 2891475, 4929660, 8155182, 13135080, 20651875, 31770970, 47923680, 70989801, 103454645, 148464520, 210155730, 293558265, 405325092, 553175000, 747508125, 999747750

Offset: 1

Author

Chai Wah Wu, Jul 26 2024

Keywords

• nonn

Comments

5-fold convolution of A000203.

Convolution of A000203 and A374977.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000385, A319083, A374951, A374977.

Programs

Maple

b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
       add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 26 2024

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, 5];
Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Jul 11 2025, after Alois P. Heinz *)

Python

from sympy import divisor_sigma
def A374978(n): return sum(divisor_sigma(j)*sum((5*divisor_sigma(i+1,3)-(5+6*i)*divisor_sigma(i+1))*(5*divisor_sigma(n-j-i-1,3)-(5+6*(n-j-i-2))*divisor_sigma(n-j-i-1)) for i in range(1,n-j-2)) for j in range(1,n))//144

Formula

a(n) = Sum_{i=1..n-1} A000203(i)*A374977(n-i).

a(n) = Sum_{i=1..n-2} A000385(i)*A374951(n-i-1).

Column k=5 of A319083.

Sum_{k=1..n} a(k) ~ Pi^10 * n^10 / 28217548800. - Vaclav Kotesovec, Sep 20 2024