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A340971 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * binomial(2*k,k).

%I A340971 #34 Aug 19 2025 09:33:36
%S A340971 1,3,33,721,23649,1032801,56317969,3682424775,280767441537,
%T A340971 24456613613401,2395993939827201,260764460901476643,
%U A340971 31213273328323059169,4075382667781540713807,576394007453263029232497
%N A340971 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * binomial(2*k,k).
%H A340971 Seiichi Manyama, <a href="/A340971/b340971.txt">Table of n, a(n) for n = 0..322</a>
%F A340971 a(n) = [x^n] 1/sqrt((1-x)*(1-(4*n+1)*x)).
%F A340971 a(n) = [x^n] (1+(2*n+1)*x+(n*x)^2)^n.
%F A340971 a(n) = n! * [x^n] exp((2*n+1)*x) * BesselI(0,2*n*x). - _Ilya Gutkovskiy_, Feb 01 2021
%F A340971 a(n) ~ exp(1/4) * 4^n * n^(n - 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Nov 13 2021
%F A340971 From _Seiichi Manyama_, Aug 19 2025: (Start)
%F A340971 a(n) = (1/4)^n * Sum_{k=0..n} (4*n+1)^k * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A340971 a(n) = Sum_{k=0..n} (-n)^k * (4*n+1)^(n-k) * binomial(n,k) * binomial(2*k,k). (End)
%t A340971 a[0] = 1; a[n_] := Sum[n^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 15, 0] (* _Amiram Eldar_, Feb 01 2021 *)
%o A340971 (PARI) a(n) = sum(k=0, n, n^k*binomial(n, k)*binomial(2*k, k));
%o A340971 (PARI) a(n) = polcoef(1/sqrt((1-x)*(1-(4*n+1)*x)+x*O(x^n)), n);
%o A340971 (PARI) a(n) = polcoef((1+(2*n+1)*x+(n*x)^2)^n, n);
%Y A340971 Main diagonal of A340970.
%Y A340971 Cf. A338979, A340972.
%K A340971 nonn
%O A340971 0,2
%A A340971 _Seiichi Manyama_, Feb 01 2021