A293295 a(n) = Sum_{k=1..n} (-1)^(n-k)*hypergeom([k, k-2-n], [], 1).

1, 5, 27, 142, 847, 5817, 45733, 405836, 4012701, 43733965, 520794991, 6726601050, 93651619867, 1398047697137, 22275111534537, 377278848390232, 6768744159489913, 128228860181918421, 2557808459478878851, 53585748788874537830, 1176328664895760953831

Offset: 1

Author

Peter Luschny, Oct 05 2017

Keywords

• nonn

Crossrefs

Cf. A003470 (n=0), A193464 (n=1), this sequence (n=2), A292898 (n>=0).

Programs

Maple

A293295 := n -> add((-1)^(n-k)*hypergeom([k, k-2-n], [], 1), k=1..n):
seq(simplify(A293295(n)), n=1..20);

Mathematica

Table[Sum[(-1)^(n-k)*HypergeometricPFQ[{k, k-2-n}, {}, 1], {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, Jul 05 2018 *)

Formula

a(n) = A292898(n, 2).

From Vaclav Kotesovec, Jul 05 2018: (Start)

Recurrence: (n^2 - 4*n + 5)*a(n) = (n^3 - 3*n^2 + 3*n + 2)*a(n-1) - (n-1)*(2*n - 3)*a(n-2) - (n^3 - 3*n^2 + 2*n + 1)*a(n-3) + (n^2 - 2*n + 2)*a(n-4).

a(n) ~ n * n!.

a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / exp(n). (End)