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

1, 3, 33, 721, 23649, 1032801, 56317969, 3682424775, 280767441537, 24456613613401, 2395993939827201, 260764460901476643, 31213273328323059169, 4075382667781540713807, 576394007453263029232497

Offset

0,2

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] exp((2*n+1)*x) * BesselI(0,2*n*x). - Ilya Gutkovskiy, Feb 01 2021

a(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

Main diagonal of A340970.

Cf. A338979, A340972.

Sequence in context: A336636 A364242 A091462 * A326328 A233319 A003715

Adjacent sequences: A340968 A340969 A340970 * A340972 A340973 A340974

Keyword

nonn

Author

Seiichi Manyama, Feb 01 2021

Status

approved