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A diagonal of triangle in A008298.
4

%I #27 Apr 24 2021 09:53:54

%S 1,9,59,450,3394,30912,293292,3032208,36290736,433762560,5925016800,

%T 83648747520,1335385128960,20323375994880,376785057196800,

%U 6493118120294400,132672192555571200,2513351450024755200,56577426980420505600,1188283280226545664000,29682641812682686464000,658094690655791972352000

%N A diagonal of triangle in A008298.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.

%H Seiichi Manyama, <a href="/A059356/b059356.txt">Table of n, a(n) for n = 2..448</a>

%F a(n) = (n-1)! * Sum_{k=1..n-1} sigma(k)*sigma(n-k)/k = (n!/2) * Sum_{k=1..n-1} sigma(k)*sigma(n-k)/(k*(n-k)). - _Seiichi Manyama_, Nov 09 2020.

%F E.g.f.: (1/2) * log( Product_{k>=1} (1 - x^k) )^2. - _Ilya Gutkovskiy_, Apr 24 2021

%t nmax = 30; Table[n!/2 * Sum[DivisorSigma[1, k] * DivisorSigma[1, n-k] / k / (n-k), {k, 1, n-1}], {n, 2, nmax}] (* _Vaclav Kotesovec_, Nov 09 2020 *)

%o (PARI) {a(n) = my(t='t); n!*polcoef(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(-t)), n), 2)} \\ _Seiichi Manyama_, Nov 07 2020

%o (PARI) {a(n)= (n-1)!*sum(k=1, n-1, sigma(k)*sigma(n-k)/k)} \\ _Seiichi Manyama_, Nov 09 2020

%o (PARI) {a(n)= n!*sum(k=1, n-1, sigma(k)*sigma(n-k)/(k*(n-k)))/2} \\ _Seiichi Manyama_, Nov 09 2020

%Y Cf. A000203, A000385, A008298.

%K nonn

%O 2,2

%A _N. J. A. Sloane_, Jan 27 2001

%E More terms from _Vladeta Jovovic_, Dec 28 2001