%I #32 Sep 19 2024 09:30:16
%S 0,0,1,9,39,120,300,645,1261,2262,3825,6160,9471,14178,20376,28965,
%T 39600,54066,71145,94248,120140,155310,193116,244560,297819,370860,
%U 443710,544554,641655,778458,904800,1085445,1248762,1483308,1688052,1991515,2244375,2626380
%N a(n) = Sum_{i+j+k=n, i,j,k >= 1} sigma(i) * sigma(j) * sigma(k).
%H Vaclav Kotesovec, <a href="/A374951/b374951.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: ( Sum_{k>=1} k * x^k/(1 - x^k) )^3 = ( Sum_{k>=1} x^k/(1 - x^k)^2 )^3.
%F a(n) = Sum_{i=1..n-2} sigma(i)*A000385(n-i-1). - _Chai Wah Wu_, Jul 25 2024
%F Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 155520. - _Vaclav Kotesovec_, Sep 19 2024
%p b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
%p add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p end:
%p a:= n-> b(n, 3):
%p seq(a(n), n=1..55); # _Alois P. Heinz_, Jul 25 2024
%o (PARI) my(N=40, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, k*x^k/(1-x^k))^3))
%o (Python)
%o from sympy import divisor_sigma
%o 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
%Y Column k=3 of A319083.
%Y Cf. A000203, A000385, A191829.
%K nonn
%O 1,4
%A _Seiichi Manyama_, Jul 25 2024