A247499 a(n) = hypergeom([1, -n, -n-1], [2], 1).

1, 2, 6, 25, 135, 896, 7042, 63841, 654901, 7491574, 94470926, 1301130777, 19423173211, 312256205652, 5376809244458, 98700795776641, 1923638785344457, 39661911384761866, 862362968121278038, 19717031047061570777, 472849461034147171791, 11866892471399392308232

Offset: 0

Author

Peter Luschny, Oct 17 2014

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Crossrefs

Row sums of A247500.

Programs

Haskell

a247499 = sum . a247500_row  -- Reinhard Zumkeller, Oct 19 2014

Maple

A247499 := n -> hypergeom([1, -n, -n-1], [2], 1):
seq(simplify(A247499(n)), n = 0..21);

Mathematica

Table[Sum[n!/k!*Binomial[n+2, k+1]/(n+2), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jul 05 2018 *)

Formula

a(n) = n!*hypergeom([-n-1], [2], -1) - 1/((n+1)*(n+2)). (original name)

a(n) = Sum_{k=0..n} (n!/k!)*binomial(n+2, k+1)/(n+2).

From Vaclav Kotesovec, Jul 05 2018: (Start)

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

a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n - 1/4) / sqrt(2). (End)

Extensions

Name updated by Peter Luschny, Jan 09 2022