A119578 a(n) = (n + n^2)*binomial(2*n,n)/2.

0, 2, 18, 120, 700, 3780, 19404, 96096, 463320, 2187900, 10161580, 46558512, 210924168, 946454600, 4212243000, 18614102400, 81746933040, 357041751660, 1551848136300, 6715600122000, 28947771052200, 124337568995640, 532337037821160, 2272426880817600, 9674281104930000

Offset: 0

Author

Zerinvary Lajos, May 31 2006

Keywords

• nonn
• easy

Comments

For n > 0, also the number of one-sided prudent walks from (0,0) to (n,n), with n+2 east steps, 2 west steps and n north steps.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..500
• Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.
• S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.

Crossrefs

Cf. A000984, A001622, A085373.

Programs

Magma

[0] cat [ (n+1)*Factorial(2*n-1)/Factorial(n-1)^2: n in [1..23] ]; // Klaus Brockhaus, Apr 30 2011

Maple

[seq ((n+n^2)*(binomial(2*n,n))/2,n=0..29)];

Mathematica

Table[(n+n^2) Binomial[2n,n]/2,{n,0,30}] (* Harvey P. Dale, Jun 02 2016 *)

Formula

a(n) = (n+1)*Gamma(2*n)/Gamma(n)^2 for n > 0. - Shanzhen Gao, Apr 26 2011

G.f.: 2 * x * (1 - x) / (1 - 4*x)^(5/2). - Ilya Gutkovskiy, Nov 17 2021

From Amiram Eldar, May 15 2022: (Start)

Sum_{n>=1} 1/a(n) = 2*Pi^2/9 - 2*Pi/sqrt(3) + 2.

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(5)*log(phi) - 8*log(phi)^2 - 2, where phi is the golden ratio (A001622). (End)

D-finite with recurrence (-n+1)*a(n) +(5*n-1)*a(n-1) +2*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jul 08 2022

a(n) = A000217(n)*A000984(n). - R. J. Mathar, Jul 08 2022