A340972 - OEIS

%I #34 Nov 13 2021 05:39:54

%S 1,-1,17,-395,13345,-592299,32630401,-2148740061,164682639745,

%T -14401797806195,1415344434226801,-154426458074411313,

%U 18523291145011712929,-2422743610992855309925,343167234011405980982625

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

%H Seiichi Manyama, <a href="/A340972/b340972.txt">Table of n, a(n) for n = 0..322</a>

%F a(n) = [x^n] 1/sqrt((1-x)*(1+(4*n-1)*x)).

%F a(n) = [x^n] (1-(2*n-1)*x+(n*x)^2)^n.

%F a(n) = n! * [x^n] BesselI(0,2*n*x) / exp((2*n-1)*x). - _Ilya Gutkovskiy_, Feb 01 2021

%F a(n) ~ (-1)^n * exp(-1/4) * 4^n * n^(n - 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Nov 13 2021

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

%o (PARI) {a(n) = polcoef(1/sqrt((1-x)*(1+(4*n-1)*x)+x*O(x^n)), n)}

%o (PARI) {a(n) = polcoef((1-(2*n-1)*x+(n*x)^2)^n, n)}

%Y Cf. A322246, A339001, A340971.

%K sign

%O 0,3

%A _Seiichi Manyama_, Feb 01 2021