A322747 - OEIS

%I #21 Dec 29 2018 03:49:07

%S 1,5,161,8749,665857,65160501,7793761249,1101696200669,

%T 179689877047297,33215554576822501,6862186181491284001,

%U 1566923219786361397005,391868347839681254572801,106523078497331434142611733,31273034455313887578671676257

%N a(n) = sqrt(1 + A322746(2*n)).

%H Seiichi Manyama, <a href="/A322747/b322747.txt">Table of n, a(n) for n = 0..296</a>

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

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

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

%F a(n) = A322790(2*n, n).

%F a(n) = T_{n}(4*n+1) where T_{n}(x) is a Chebyshev polynomial of the first kind.

%F a(n) ~ exp(1/4) * 2^(3*n - 1) * n^n. - _Vaclav Kotesovec_, Dec 25 2018

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

%o (PARI) {a(n) = polchebyshev(n, 1, 4*n+1)}

%Y Cf. A322746, A322790.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 25 2018