%I #52 Jan 30 2020 21:29:14

%S 0,0,1,2,7,24,89,342,1355,5492,22669,94962,402703,1725424,7458065,

%T 32482798,142414867,628037612,2783922197,12397342698,55436525591,

%U 248819728360,1120584933401,5062273384422,22933667676187

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

%C 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

%H Muniru A Asiru, <a href="/A052705/b052705.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=660">Encyclopedia of Combinatorial Structures 660</a>

%H C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.

%H D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.4153/CJM-1997-015-x">On some alternative characterizations of Riordan arrays, Can. J. Math., 49 (2) (1997) 301-310.

%F 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.

%F a(n+2) = Sum_{k=0..n} Sum_{j=0..n} C(j,n-j)*A001263(j,k). - _Paul Barry_, Jun 30 2009

%F 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

%F 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

%F 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

%F a(n) ~ 2^(n + 3/4) * (1 + sqrt(2))^(n - 5/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 03 2019

%p 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);

%t CoefficientList[Series[(2x^2)/(1-2x-2x^2+Sqrt[1-4x-4x^2]),{x,0,30}],x] (* _Harvey P. Dale_, Dec 16 2014 *)

%t 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 *)

%o (Maxima)

%o a(n):=(sum(binomial(2*n-2*j,n)*binomial(n,j-1)/(n-j),j,1,n/2)); /* _Vladimir Kruchinin_, Jan 16 2015 */

%Y Row sums of A071945, cf. A000108.

%K easy,nonn

%O 0,4

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

%E More terms from _Emeric Deutsch_, Mar 07 2004