A181768 G.f.: (1/2)*(3 - sqrt((1-5*x)/(1-x))).

1, 1, 2, 5, 15, 51, 188, 731, 2950, 12235, 51822, 223191, 974427, 4302645, 19181100, 86211885, 390248055, 1777495635, 8140539950, 37463689775, 173164232965, 803539474345, 3741930523740, 17481709707825, 81912506777200, 384847173838501, 1812610804416698, 8556895079642921, 40480850291739165, 191884148712996795, 911225151259732188, 4334673398737025619, 20653004146207902678, 98551406393189773875

Offset: 0

Author

N. J. A. Sloane, Nov 12 2010

Keywords

• nonn

Comments

Same as A007317 if the first 1 is omitted. Has several combinatorial interpretations so deserves its own entry.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Ira M. Gessel, Jang Soo Kim, A note on 2-distant noncrossing partitions and weighted Motzkin paths, arXiv:1003.5301 [math.CO], 2010.
• Ira M. Gessel, Jang Soo Kim, A note on 2-distant noncrossing partitions and weighted Motzkin paths, Discrete Math. 310 (2010), no. 23, 3421-3425.

Crossrefs

Cf. A007317.

Programs

Maple

A181768 := n -> `if`(n=0, 1, JacobiP(n-1,1,-n-1/2,9)/n):
seq(round(evalf(A181768(n),99)), n=0..33); # Peter Luschny, Sep 23 2014

Mathematica

CoefficientList[Series[3/2-Sqrt[(1-5x)/(1-x)]/2,{x,0,40}],x] (* Harvey P. Dale, Jul 28 2013 *)

PARI

x='x + O('x^50); Vec((1/2)*(3 - sqrt((1-5*x)/(1-x)))) \\ G. C. Greubel, Feb 12 2017

Formula

a(n) ~ 5^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013

D-finite with recurrence: n*a(n) +2*(-3*n+4)*a(n-1) +5*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 06 2013

a(n) = JacobiP(n-1,1,-n-1/2,9)/n for n>0. - Peter Luschny, Sep 23 2014

a(n) = 1 + Sum_{k=1..n-1} a(k) * a(n-k). - Ilya Gutkovskiy, Jul 01 2020