A294117 a(n) = (n!)^2 * Sum_{k=1..n} binomial(n,k) / k^2.

1, 9, 139, 3460, 129076, 6831216, 492314544, 46810296576, 5724123883776, 881047053849600, 167511790501401600, 38685942660873830400, 10689310289146278297600, 3485920800452969462169600, 1325434521073620201431040000, 581241452210335678204477440000

Offset: 1

Author

Vaclav Kotesovec, Oct 23 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..238

Crossrefs

Cf. A000424, A060237, A103213.

Programs

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

Maple

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):
map(f, [$1..20]); # Robert Israel, Oct 23 2017

Mathematica

Table[n!^2*Sum[Binomial[n, k]/k^2, {k, 1, n}], {n, 1, 20}]
Table[n!^2*n*HypergeometricPFQ[{1, 1, 1, 1 - n}, {2, 2, 2}, -1], {n, 1, 20}]

Formula

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(n) ~ (n!)^2 * 2^(n+2) / n^2.