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%I #11 Sep 08 2022 08:46:20
%S 1,9,139,3460,129076,6831216,492314544,46810296576,5724123883776,
%T 881047053849600,167511790501401600,38685942660873830400,
%U 10689310289146278297600,3485920800452969462169600,1325434521073620201431040000,581241452210335678204477440000
%N a(n) = (n!)^2 * Sum_{k=1..n} binomial(n,k) / k^2.
%H Robert Israel, <a href="/A294117/b294117.txt">Table of n, a(n) for n = 1..238</a>
%F a(n) = (5*n^2 - 7*n + 3)*a(n-1) - (n-1)^2*(9*n^2 - 24*n + 17)*a(n-2) + (n-2)^3*(n-1)^2*(7*n - 13)*a(n-3) - 2*(n-3)^3*(n-2)^3*(n-1)^2*a(n-4).
%F a(n) ~ (n!)^2 * 2^(n+2) / n^2.
%p f:= gfun:-rectoproc({a(n) = (5*n^2 - 7*n + 3)*a(n-1) - (n-1)^2*(9*n^2 - 24*n + 17)*a(n-2) + (n-2)^3*(n-1)^2*(7*n - 13)*a(n-3) - 2*(n-3)^3*(n-2)^3*(n-1)^2*a(n-4),a(1)=1,a(2)=9,a(3)=139,a(4)=3460},a(n),remember):
%p map(f, [$1..20]); # _Robert Israel_, Oct 23 2017
%t Table[n!^2*Sum[Binomial[n, k]/k^2, {k, 1, n}], {n, 1, 20}]
%t Table[n!^2*n*HypergeometricPFQ[{1, 1, 1, 1 - n}, {2, 2, 2}, -1], {n, 1, 20}]
%o (Magma) I:=[1,9,139,3460]; [n le 4 select I[n] else (5*n^2- 7*n+3)*Self(n-1)-(n-1)^2*(9*n^2-24*n+17)*Self(n-2)+(n-2)^3*(n-1)^2*(7*n-13)*Self(n-3)-2*(n-3)^3*(n-2)^3*(n-1)^2*Self(n-4): n in [1..16]]; // _Vincenzo Librandi_, Oct 24 2017
%Y Cf. A000424, A060237, A103213.
%K nonn
%O 1,2
%A _Vaclav Kotesovec_, Oct 23 2017
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