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A340972 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k) * binomial(2*k,k).

%I A340972 #40 Aug 19 2025 22:56:47
%S A340972 1,-1,17,-395,13345,-592299,32630401,-2148740061,164682639745,
%T A340972 -14401797806195,1415344434226801,-154426458074411313,
%U A340972 18523291145011712929,-2422743610992855309925,343167234011405980982625
%N A340972 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k) * binomial(2*k,k).
%H A340972 Seiichi Manyama, <a href="/A340972/b340972.txt">Table of n, a(n) for n = 0..322</a>
%F A340972 a(n) = [x^n] 1/sqrt((1-x)*(1+(4*n-1)*x)).
%F A340972 a(n) = [x^n] (1-(2*n-1)*x+(n*x)^2)^n.
%F A340972 a(n) = n! * [x^n] BesselI(0,2*n*x) / exp((2*n-1)*x). - _Ilya Gutkovskiy_, Feb 01 2021
%F A340972 a(n) ~ (-1)^n * exp(-1/4) * 4^n * n^(n - 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Nov 13 2021
%F A340972 From _Seiichi Manyama_, Aug 19 2025: (Start)
%F A340972 a(n) = (1/4)^n * Sum_{k=0..n} (-4*n+1)^k * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A340972 a(n) = Sum_{k=0..n} n^k * (-4*n+1)^(n-k) * binomial(n,k) * binomial(2*k,k). (End)
%t A340972 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 A340972 (PARI) a(n) = sum(k=0, n, (-n)^k*binomial(n, k)*binomial(2*k, k));
%o A340972 (PARI) a(n) = polcoef(1/sqrt((1-x)*(1+(4*n-1)*x)+x*O(x^n)), n);
%o A340972 (PARI) a(n) = polcoef((1-(2*n-1)*x+(n*x)^2)^n, n);
%Y A340972 Cf. A322246, A339001, A340971.
%K A340972 sign
%O A340972 0,3
%A A340972 _Seiichi Manyama_, Feb 01 2021