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

1, -1, 17, -395, 13345, -592299, 32630401, -2148740061, 164682639745, -14401797806195, 1415344434226801, -154426458074411313, 18523291145011712929, -2422743610992855309925, 343167234011405980982625

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..322

Formula

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

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

a(n) = n! * [x^n] BesselI(0,2*n*x) / exp((2*n-1)*x). - Ilya Gutkovskiy, Feb 01 2021

a(n) ~ (-1)^n * exp(-1/4) * 4^n * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Nov 13 2021

Mathematica

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 *)

Prog

(PARI) {a(n) = sum(k=0, n, (-n)^k*binomial(n, k)*binomial(2*k, k))}
(PARI) {a(n) = polcoef(1/sqrt((1-x)*(1+(4*n-1)*x)+x*O(x^n)), n)}
(PARI) {a(n) = polcoef((1-(2*n-1)*x+(n*x)^2)^n, n)}

Crossrefs

Cf. A322246, A339001, A340971.

Sequence in context: A350926 A327732 A012200 * A340353 A007925 A097201

Adjacent sequences: A340969 A340970 A340971 * A340973 A340974 A340975

Keyword

sign

Author

Seiichi Manyama, Feb 01 2021

Status

approved