A323208 a(n) = hypergeometric([-n - 1, n + 2], [-n - 2], n).

1, 5, 67, 1606, 55797, 2537781, 142648495, 9549411950, 741894295369, 65620725560578, 6511108452179611, 716273662860469000, 86527644431076024637, 11387523335268377432565, 1621766490238904658104583, 248507974510512755641561366, 40769019250019155227631614225

Offset: 0

Author

Peter Luschny, Feb 25 2019

Keywords

• nonn

Crossrefs

Cf. A323206, A238762.

Programs

Maple

# The function ballot is defined in A238762.
a := n -> add(ballot(2*j, 2*n+2)*n^j, j=0..n+1):
seq(a(n), n=0..16);

Mathematica

a[n_] := Hypergeometric2F1[-n - 1, n + 2, -n - 2, n];
Table[a[n], {n, 0, 16}]

Formula

a(n) = A323206(n, n+1).

a(n) = Sum_{j=0..n+1} (binomial(2*(n+1)-j,n+1)-binomial(2*(n+1)-j,n+2))*n^(n+1-j).

a(n) = Sum_{j=0..n+1} binomial(n+1+j, n+1)*(1 - j/(n+2))*n^j.

a(n) = 1 + Sum_{j=0..n} ((1+j)*binomial(2*(n+1)-j, n+2)/(n+1-j))*n^(n+1-j).

a(n) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^(n+1))/(1+(n-1)*x), n>0.

a(n) ~ (4^(n + 2)*n^(n + 3))/(sqrt(Pi)*(1 - 2*n)^2*(n + 1)^(3/2)).