A322746 a(n) = 1/2 * (-1 + Sum_{k=0..n} binomial(2*n,2*k)*(n+1)^(n-k)*n^k).

0, 1, 24, 675, 25920, 1275125, 76545000, 5425069447, 443365544448, 41047124680809, 4245890890571000, 485307363135371051, 60742714406414040000, 8262695239025750162653, 1213734518568509516047560, 191478489107270936785743375, 32288451913272713227175006208

Offset: 0

Author

Seiichi Manyama, Dec 25 2018

Keywords

• nonn

Examples

			(sqrt(3) + sqrt(2))^2 = 5 + 2*sqrt(6) = sqrt(25) + sqrt(24). So a(2) = 24.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..321
• Wikipedia, Chebyshev polynomials.
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Main diagonal of A322699.

Cf. A322747.

Programs

PARI

{a(n) = 1/2*(-1+sum(k=0, n, binomial(2*n,2*k)*(n+1)^(n-k)*n^k))}

PARI

{a(n) = (polchebyshev(n, 1, 2*n+1)-1)/2}

Formula

sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^n.

sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^n.

a(n) = (A173174(n) - 1)/2.

a(n) ~ exp(1/2) * 2^(2*n - 2) * n^n. - Vaclav Kotesovec, Dec 25 2018