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