A342207 - OEIS

%I #22 Mar 06 2021 01:46:45

%S 1,4,35,496,9701,241956,7338631,262184896,10783446409,501827040100,

%T 26069206375211,1495427735314800,93885489910449901,

%U 6403169506981578436,471427031236487965199,37265225545829174607616,3147895910861898495432209

%N a(n) = U(n,n+1) where U(n,x) is a Chebyshev polynomial of the second kind.

%H Seiichi Manyama, <a href="/A342207/b342207.txt">Table of n, a(n) for n = 0..351</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

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

%F a(n) ~ exp(1) * 2^n * n^n. - _Vaclav Kotesovec_, Mar 05 2021

%t Table[ChebyshevU[n, n + 1], {n, 0, 16}] (* _Amiram Eldar_, Mar 05 2021 *)

%o (PARI) a(n) = polchebyshev(n, 2, n+1);

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

%o (PARI) a(n) = sum(k=0, n, (2*n)^k*binomial(n+1+k, 2*k+1));

%Y Cf. A107995, A323118, A342168, A342205.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 05 2021