A052705 Expansion of 2*x^2/(1 - 2*x - 2*x^2 + sqrt(1 - 4*x - 4*x^2)).

0, 0, 1, 2, 7, 24, 89, 342, 1355, 5492, 22669, 94962, 402703, 1725424, 7458065, 32482798, 142414867, 628037612, 2783922197, 12397342698, 55436525591, 248819728360, 1120584933401, 5062273384422, 22933667676187

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Number of underdiagonal paths from (0,0) to the line x=n-2, using only steps R=(1,0), V=(0,1) and D=(2,1). E.g., a(4)=7 because we have RR, RRV, RVR, D, RVRV, RRVV and DV. - Emeric Deutsch, Dec 21 2003

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 660
• C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
• D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Can. J. Math., 49 (2) (1997) 301-310.

Crossrefs

Row sums of A071945, cf. A000108.

Programs

Maple

spec := [S,{S=Prod(B,B),C=Prod(S,Z),B=Union(S,C,Z)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

CoefficientList[Series[(2x^2)/(1-2x-2x^2+Sqrt[1-4x-4x^2]),{x,0,30}],x] (* Harvey P. Dale, Dec 16 2014 *)
Join[{0,0},Table[(Binomial[2(m-1),m]HypergeometricPFQ[{(2-m)/2,(3-m)/2,-m},{3/2-m,2-m},-1])/(m-1),{m,2,20}]] (* Benedict W. J. Irwin, Sep 13 2016 *)

Maxima

a(n):=(sum(binomial(2*n-2*j,n)*binomial(n,j-1)/(n-j),j,1,n/2)); /* Vladimir Kruchinin, Jan 16 2015 */

Formula

D-finite with recurrence: a(1)=0, a(2)=1, a(3)=2, 4*(n+1)*a(n) + (10+8*n)*a(n+1) + (2+3*n)*a(n+2) + (-n-3)*a(n+3) = 0.

a(n+2) = Sum_{k=0..n} Sum_{j=0..n} C(j,n-j)*A001263(j,k). - Paul Barry, Jun 30 2009

a(n) = Sum_{j=1..floor(n/2)} C(2*n-2*j,n)*C(n,j-1)/(n-j). - Vladimir Kruchinin, Jan 16 2015

G.f.: A(x) satisfies A(x) = C(x*(1+A(x)))^2, where x*C(x) is g.f. of Catalan numbers. - Vladimir Kruchinin, Jan 16 2015

a(n) = C(2*n-2,n)*3F2((2-n)/2,(3-n)/2,-n;3/2-n,2-n;-1)/(n-1), n>1. - Benedict W. J. Irwin, Sep 13 2016

a(n) ~ 2^(n + 3/4) * (1 + sqrt(2))^(n - 5/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019

Extensions

More terms from Emeric Deutsch, Mar 07 2004