A279013 a(n) = Sum_{k=0..n} binomial(2*k,k)/(k+1)*binomial(2*n-1,n-k).

1, 2, 8, 35, 161, 768, 3773, 19006, 97840, 513264, 2737121, 14805805, 81082383, 448805300, 2507310567, 14120503129, 80082573017, 456977964520, 2621830478785, 15114658956625, 87508451311125, 508589225952740, 2966098696204660

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f.: (2*(1-sqrt(1-((1-sqrt(1-4*x))^2)/x))*(1/sqrt(1-4*x)+1)/2*x)/((1-sqrt(1-4*x))^2).

Conjecture D-finite with recurrence: 2*n*(n+1)*(2*n-3)*a(n) -n*(101*n^2-312*n+203)*a(n-1) +(995*n^3-5570*n^2+9567*n-4984)*a(n-2) +2*(-2393*n^3+19300*n^2-50494*n+42835)*a(n-3) +4*(2*n7)*(1408*n^2-9889*n+16690)*a(n-4) -2600*(n-5)*(2*n-7)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Jan 27 2020

a(n) ~ 5^(2*n + 1/2) / (3^(3/2) * sqrt(Pi) * n^(3/2) * 2^(2*n - 2)). - Vaclav Kotesovec, Nov 19 2021

Mathematica

Table[Sum[Binomial[2k, k]/(k+1) Binomial[2n-1, n-k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Feb 06 2019 *)

Prog

(Maxima)
taylor((2*(1-sqrt(1-((1-sqrt(1-4*x))^2)/x))*(1/sqrt(1-4*x)+1)/2*x)/((1-sqrt(1-4*x))^2), x, 0, 27);

Crossrefs

Cf. A000108

Sequence in context: A184786 A082759 A243204 * A137265 A364472 A303070

Adjacent sequences: A279010 A279011 A279012 * A279014 A279015 A279016

Keyword

nonn

Author

Vladimir Kruchinin, Dec 03 2016

Status

approved