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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).

%I A374978 #29 Jul 12 2025 08:31:39
%S A374978 0,0,0,0,1,15,110,545,2095,6713,18750,47040,108185,231640,467034,
%T A374978 894605,1639680,2891475,4929660,8155182,13135080,20651875,31770970,
%U A374978 47923680,70989801,103454645,148464520,210155730,293558265,405325092,553175000,747508125,999747750
%N 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).
%C A374978 5-fold convolution of A000203.
%C A374978 Convolution of A000203 and A374977.
%H A374978 Robert Israel, <a href="/A374978/b374978.txt">Table of n, a(n) for n = 1..10000</a>
%F A374978 a(n) = Sum_{i=1..n-1} A000203(i)*A374977(n-i).
%F A374978 a(n) = Sum_{i=1..n-2} A000385(i)*A374951(n-i-1).
%F A374978 Column k=5 of A319083.
%F A374978 Sum_{k=1..n} a(k) ~ Pi^10 * n^10 / 28217548800. - _Vaclav Kotesovec_, Sep 20 2024
%p A374978 b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p A374978       `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
%p A374978        add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A374978     end:
%p A374978 a:= n-> b(n, 5):
%p A374978 seq(a(n), n=1..55);  # _Alois P. Heinz_, Jul 26 2024
%t A374978 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]]]];
%t A374978 a[n_] := b[n, 5];
%t A374978 Table[a[n], {n, 1, 55}] (* _Jean-François Alcover_, Jul 11 2025, after _Alois P. Heinz_ *)
%o A374978 (Python)
%o A374978 from sympy import divisor_sigma
%o A374978 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
%Y A374978 Cf. A000203, A000385, A319083, A374951, A374977.
%K A374978 nonn
%O A374978 1,6
%A A374978 _Chai Wah Wu_, Jul 26 2024