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A323012 a(n) = (1/sqrt(n^2+1)) * T_{2*n+1}(sqrt(n^2+1)) where T_{n}(x) is a Chebyshev polynomial of the first kind.

%I A323012 #25 Sep 08 2022 08:46:23
%S A323012 1,5,305,53353,18674305,10928351501,9616792908241,11868363584907985,
%T A323012 19553538801258341377,41456387654578883552149,
%U A323012 109939727677547706703222001,356521758767660233608385698361,1387930545993760882531890016305025
%N A323012 a(n) = (1/sqrt(n^2+1)) * T_{2*n+1}(sqrt(n^2+1)) where T_{n}(x) is a Chebyshev polynomial of the first kind.
%H A323012 Seiichi Manyama, <a href="/A323012/b323012.txt">Table of n, a(n) for n = 0..193</a>
%H A323012 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%H A323012 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A323012 a(n) = Sum_{k=0..n} binomial(2*n+1,2*k)*(n^2+1)^(n-k)*n^(2*k).
%F A323012 a(n) ~ 2^(2*n) * n^(2*n). - _Vaclav Kotesovec_, Jan 02 2019
%t A323012 Join[{1}, Table[Sum[Binomial[2 n + 1, 2 k] (n^2 + 1)^(n - k) n^(2 k), {k, 0, n}], {n, 20}]] (* _Vincenzo Librandi_, Jan 03 2019 *)
%o A323012 (PARI) {a(n) = sum(k=0, n, binomial(2*n+1, 2*k)*(n^2+1)^(n-k)*n^(2*k))}
%o A323012 (Magma) [&+[Binomial(2*n+1,2*k)*(n^2+1)^(n-k)*n^(2*k): k in [0..n]]: n in [0..15]]; // _Vincenzo Librandi_, Jan 03 2019
%Y A323012 Diagonal of A188647.
%K A323012 nonn
%O A323012 0,2
%A A323012 _Seiichi Manyama_, Jan 02 2019