A051133 a(n) = binomial(2n,n)*n*(2n+1)/2.

0, 3, 30, 210, 1260, 6930, 36036, 180180, 875160, 4157010, 19399380, 89237148, 405623400, 1825305300, 8143669800, 36064823400, 158685222960, 694247850450, 3022020054900, 13095420237900, 56517076816200, 243023430309660, 1041528987041400

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = (1/2) * A000911(n-1).

a(n) = (1/2)*A000984(n+1)*A000217(n). - Zerinvary Lajos, May 05 2007

a(n) = 3*A002802(n-1). - Zerinvary Lajos, Jun 02 2007

(-n+1)*a(n) + 2*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013

G.f.: 3*x * (1 - 4*x)^(-5/2). - Michael Somos, Sep 09 2013

Sum_{n>=1} 1/a(n) = 4 - 2*Pi/sqrt(3). - Amiram Eldar, Oct 22 2020

Example

G.f. = 3*x + 30*x^2 + 210*x^3 + 1260*x^4 + 6930*x^5 + 36036*x^6 + ...

Maple

seq(binomial(2*n, n)*binomial(n, (n-2))/2, n=1..23); # Zerinvary Lajos, May 05 2007

Mathematica

a[ n_]:= SeriesCoefficient[ 3x(1-4x)^(-5/2), {x, 0, n}]; (* Michael Somos, Sep 09 2013 *)
Table[Binomial[2*n, n]*n*(2*n + 1)/2, {n, 0, 22}] (* Amiram Eldar, Oct 22 2020 *)

Prog

(PARI) {a(n) = if( n<1, 0, (2*n + 1)! / (2 * n! *(n-1)!))}; /* Michael Somos, Sep 09 2013 */
(PARI) {a(n) = 2^(n+2) * polcoeff( pollegendre( n+3), n-1)}; /* Michael Somos, Sep 09 2013 */
(Magma) [Binomial(2*n, n)*n*(2*n+1)/2: n in [0..25]]; // G. C. Greubel, Feb 10 2019
(Sage) [binomial(2*n, n)*n*(2*n+1)/2 for n in (0..25)] # G. C. Greubel, Feb 10 2019

Crossrefs

Cf. A000911, A000984, A000217, A002802.

Sequence in context: A121100 A203366 A130546 * A332426 A043030 A178015

Adjacent sequences: A051130 A051131 A051132 * A051134 A051135 A051136

Keyword

nonn,easy

Author

Wouter Meeussen

Status

approved