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A374979 a(n) = Sum_{i+j+k+l+m+r=n, i,j,k,l,m,r >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l)*sigma(m)*sigma(r).

%I A374979 #25 Sep 20 2024 06:22:29
%S A374979 0,0,0,0,0,1,18,159,942,4281,16050,51932,149532,391524,947246,2143677,
%T A374979 4581204,9316195,18138636,33984912,61534652,108055425,184582014,
%U A374979 307515038,500798058,798762453,1249917936,1921788036,2907159804,4332046200,6365441400,9232216725
%N A374979 a(n) = Sum_{i+j+k+l+m+r=n, i,j,k,l,m,r >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l)*sigma(m)*sigma(r).
%C A374979 6-fold convolution of A000203.
%C A374979 Convolution of A000203 and A374978.
%C A374979 a(n) = Sum_{i=1..n-1} A000203(i)*A374978(n-i).
%C A374979 a(n) = Sum_{i=1..n-2} A000385(i)*A374977(n-i-1).
%C A374979 a(n) = Sum_{i=1..n-1} A374951(i)*A374951(n-i).
%C A374979 a(n) = Sum_{i+j+k=n-3, i,j,k>=1} A000385(i)*A000385(j)*A000385(k).
%C A374979 Column k=6 of A319083.
%C A374979 In general, if the sequence "a" is a k-fold convolution of A000203, then Sum_{k=1..n} a(k) ~ Pi^(2*k) * n^(2*k) / (6^k * (2*k)!). - _Vaclav Kotesovec_, Sep 20 2024
%H A374979 Alois P. Heinz, <a href="/A374979/b374979.txt">Table of n, a(n) for n = 1..10000</a>
%F A374979 Sum_{k=1..n} a(k) ~ Pi^12 * n^12 / 22348298649600. - _Vaclav Kotesovec_, Sep 20 2024
%p A374979 b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p A374979       `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
%p A374979        add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A374979     end:
%p A374979 a:= n-> b(n, 6):
%p A374979 seq(a(n), n=1..55);  # _Alois P. Heinz_, Jul 26 2024
%o A374979 (Python)
%o A374979 from functools import lru_cache
%o A374979 from sympy import divisor_sigma
%o A374979 def A374979(n):
%o A374979     @lru_cache(maxsize=None)
%o A374979     def g(x):
%o A374979         f = factorint(x+1).items()
%o A374979         return(5*prod((p**(3*(e+1))-1)//(p**3-1) for p,e in f)-(5+6*x)*prod((p**(e+1)-1)//(p-1) for p, e in f))//12
%o A374979     return sum(g(i)*g(j)*g(n-3-i-j) for i in range(1,n-4) for j in range(1,n-i-3))
%Y A374979 Cf. A000203, A000385, A319083, A374951, A374977, A374978.
%K A374979 nonn
%O A374979 1,7
%A A374979 _Chai Wah Wu_, Jul 26 2024