A323118 a(n) = U_{n}(n) where U_{n}(x) is a Chebyshev polynomial of the second kind.

1, 2, 15, 204, 3905, 96030, 2883167, 102213944, 4178507265, 193501094490, 10011386405999, 572335117886532, 35827847605137601, 2437406399741075126, 179059769134174484415, 14127079203550978667760, 1191321539697176278429697, 106935795565608726499866930

Offset: 0

Author

Seiichi Manyama, Jan 05 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..351
• Wikipedia, Chebyshev polynomials.

Crossrefs

Main diagonal of A323182.

Cf. A115066, A318192, A323117, A349073, A349074, A349075, A349076.

Programs

Mathematica

Table[ChebyshevU[n, n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)

PARI

a(n) = polchebyshev(n, 2, n);

PARI

a(n) = sum(k=0, n\2, (n^2-1)^k*n^(n-2*k)*binomial(n+1, 2*k+1));

PARI

a(n) = sum(k=0, n, (2*n-2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 03 2021

Formula

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

a(n) ~ 2^n * n^n. - Vaclav Kotesovec, Jan 05 2019

a(n) = Sum_{k=0..n} (2*n-2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (2*n-2)^k * binomial(n+1+k,2*k+1). - Seiichi Manyama, Mar 03 2021