A374951 a(n) = Sum_{i+j+k=n, i,j,k >= 1} sigma(i) * sigma(j) * sigma(k).

0, 0, 1, 9, 39, 120, 300, 645, 1261, 2262, 3825, 6160, 9471, 14178, 20376, 28965, 39600, 54066, 71145, 94248, 120140, 155310, 193116, 244560, 297819, 370860, 443710, 544554, 641655, 778458, 904800, 1085445, 1248762, 1483308, 1688052, 1991515, 2244375, 2626380

Offset: 1

Author

Seiichi Manyama, Jul 25 2024

Keywords

• nonn

Links

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

Crossrefs

Column k=3 of A319083.

Cf. A000203, A000385, A191829.

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, 3):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 25 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, 3];
Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Mar 14 2025, after Alois P. Heinz *)

PARI

my(N=40, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, k*x^k/(1-x^k))^3))

Python

from sympy import divisor_sigma
def A374951(n): return (60*sum(divisor_sigma(i)*divisor_sigma(n-i,3) for i in range(1,n))+divisor_sigma(n)*(9*n*(2*n-1)+1)-5*divisor_sigma(n,3)*(3*n-1))//144  # Chai Wah Wu, Jul 25 2024

Formula

G.f.: ( Sum_{k>=1} k * x^k/(1 - x^k) )^3 = ( Sum_{k>=1} x^k/(1 - x^k)^2 )^3.

a(n) = Sum_{i=1..n-2} sigma(i)*A000385(n-i-1). - Chai Wah Wu, Jul 25 2024

Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 155520. - Vaclav Kotesovec, Sep 19 2024