A278429 a(n) = Sum_{k=0..n} binomial(k+n-2,k)*binomial(2*n+1,k+n+1).

1, 3, 16, 102, 699, 4973, 36194, 267480, 1998565, 15057255, 114179652, 870351386, 6662847871, 51189449457, 394476780694, 3047878296556, 23602623675273, 183142111511819, 1423578146798168, 11082963785614926, 86405502413568259

Offset

0,2

Links

Indranil Ghosh, Table of n, a(n) for n = 0..500

Formula

G.f.: x*(1-2*x*C(2*x))/sqrt(1-8*x)/(x*C(2*x))/(1-x*C(2*x))^3, where C(x) is g.f. of Catalan numbers.

a(n) = binomial(2n+1, n+1) * 2F1(n-1, -n; n+2; -1). - Jean-François Alcover, Nov 22 2016

a(n) ~ 2^(3*n+4)/(27*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 22 2016

D-finite with recurrence n*a(n) +(-5*n+2)*a(n-1) +6*(n-14)*a(n-2) +4*(-53*n+240)*a(n-3) +112*(-2*n+5)*a(n-4)=0. - R. J. Mathar, Feb 08 2021

Mathematica

Table[Binomial[2*n+1, n+1]*Hypergeometric2F1[n-1, -n, n+2, -1], {n, 0, 20}] (* Jean-François Alcover, Nov 22 2016 *)

Prog

(Maxima)
C(x):=(1-sqrt(1-4*x))/(2*x);
taylor(x/sqrt(1-8*x)/(x*C(2*x))/(1-x*C(2*x))^3*(1-2*x*C(2*x)), x, 0, 10);
(Magma) m:=30; [&+[Binomial(k+n-2, k)*Binomial(2*n+1, k+n+1): k in [0..m]]: n in [0..30]]; // Vincenzo Librandi, Nov 22 2016
(PARI) a(n) = sum(k=0, n, binomial(k+n-2, k)*binomial(2*n+1, k+n+1)); \\ Indranil Ghosh, Mar 03 2017

Crossrefs

Cf. A000108.

Sequence in context: A009007 A000949 A091637 * A341320 A365752 A207434

Adjacent sequences: A278426 A278427 A278428 * A278430 A278431 A278432

Keyword

nonn

Author

Vladimir Kruchinin, Nov 22 2016

Status

approved