A242369 a(n) = P(n, 1, -2*n-1, 1-2*n)/(n+1), P the Jacobi polynomial.

1, 1, 3, 19, 185, 2426, 39907, 788019, 18130401, 475697854, 14004694451, 456820603086, 16343563014649, 636020474595988, 26736885607750515, 1207031709414024451, 58225055056545820545, 2988064457570991780854, 162517551565531508113699, 9336340704734213892357498

Offset: 0

Author

Peter Luschny, Jun 08 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350

Crossrefs

Main diagonal of A008550, A243631.

Cf. A204057.

Programs

Magma

A242369:= func< n | n eq 0 select 1 else (&+[Binomial(n, j)^2*(n-j)*n^(j-1)/(j+1): j in [0..n-1]]) >;
[A242369(n): n in [0..30]]; // G. C. Greubel, Feb 16 2021

Maple

a := n -> `if`(n=0,1, add(binomial(n,j)^2*(n-j)/(j+1)*n^(j-1), j=0..n-1)); seq(a(n), n=0..20);

Mathematica

Table[JacobiP[n, 1, -2*n-1, 1-2*n]/(n+1), {n, 0, 20}]

Sage

def A242369(n): return 1 if n==0 else sum( binomial(n, j)^2*(n-j)*n^(j-1)/(j+1) for j in [0..n-1])
[A242369(n) for n in [0..20]] # G. C. Greubel, Feb 16 2021

Formula

a(n) = 2F1([1-n, -n], [2], n), 2F1 the hypergeometric function.

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

a(n) ~ (sqrt(n)+1)^(2*n+1)/(2*sqrt(Pi)*(n+1/2)^(9/4)). - Peter Luschny, Nov 17 2014