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%I #28 Sep 08 2022 08:45:03
%S 1,7,85,1660,48076,1942416,104587344,7245893376,628308907776,
%T 66687811660800,8506654697548800,1284292319599411200,
%U 226530955276874956800,46165213716463676620800
%N a(n) = n!^2 * Sum_{m=1..n}( Sum_{k=1..m} 1/(k*m) ).
%H G. C. Greubel, <a href="/A060237/b060237.txt">Table of n, a(n) for n = 1..252</a>
%F a(n) = a(n-1) * n^2 + (n-1)! *n! * Sum_{k=1..n} 1/k.
%F From _Vladeta Jovovic_, Jan 29 2005: (Start)
%F Sum_{n>=0} a(n)*x^n/n!^2 = -dilog(1/(1-x))/(1-x).
%F a(n) = n!^2*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k^2. (End)
%F From _Vaclav Kotesovec_, Oct 23 2017: (Start)
%F a(n) = (3*n^2 - 3*n + 1)*a(n-1) - 3*(n-1)^4*a(n-2) + (n-2)^3*(n-1)^3*a(n-3).
%F a(n) ~ n!^2 * log(n)^2/2 * (1 + 2*gamma/log(n) + (Pi^2/6 + gamma^2)/log(n)^2), where gamma is the Euler-Mascheroni constant (A001620). (End)
%e a(2) = 2!^2 *(1/(1*1) + 1/(1*2) + 1/(2*2)) = 7.
%t Table[n!^2*Sum[(-1)^(k+1)*Binomial[n,k]/k^2, {k,1,n}], {n,1,30}] (* or *) Table[n!^2*Sum[Sum[1/(k*m), {k,1,m}], {m,1,n}], {n,1,30}](* _G. C. Greubel_, Aug 30 2018 *)
%o (PARI) for(n=1,30, print1(n!^2*sum(k=1,n, (-1)^(k+1)*binomial(n,k)/k^2), ", ")) \\ _G. C. Greubel_, Aug 30 2018
%o (Magma) [Factorial(n)^2*(&+[(-1)^(k+1)*Binomial(n,k)/k^2: k in [1..n]]): n in [1..30]]; // _G. C. Greubel_, Aug 30 2018
%Y Essentially the same as A000424.
%K easy,nonn
%O 1,2
%A _Leroy Quet_, Mar 21 2001
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