A323217 a(n) = hypergeometric([-n, n + 1], [-n - 1], n + 1).

1, 3, 25, 413, 10746, 387607, 17981769, 1022586105, 68964092542, 5384626548491, 477951767068986, 47546350648784341, 5240644323742274500, 634033030117301108127, 83540992651137240168361, 11908866726507685451458545

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)*(n+1)^j, j=0..n):
seq(a(n), n=0..16);

Mathematica

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

Formula

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

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

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

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

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

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