A323209 a(n) = hypergeometric([-n, n + 1], [-n - 1], n).

1, 2, 13, 190, 4641, 161376, 7312789, 409186310, 27272680705, 2110472708140, 186023930383501, 18401769878685172, 2018938571514794593, 243319689384354960300, 31955654188732155634341, 4542582850906442990797126, 694922224386422689648830465

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*k, 2*n)*n^k, k=0..n):
seq(a(n), n=0..16);

Mathematica

a[n_] := Hypergeometric2F1[-n, n + 1, -n - 1, n];
Table[a[n], {n, 0, 14}]

Formula

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

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

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

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

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

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