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

0,4

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.

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

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 *)

Prog

(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 */

Crossrefs

Row sums of A071945, cf. A000108.

Sequence in context: A151293 A122446 A150390 * A150391 A150392 A150393

Adjacent sequences: A052702 A052703 A052704 * A052706 A052707 A052708

Keyword

easy,nonn

Author

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

Extensions

More terms from _Emeric Deutsch_, Mar 07 2004

Status

approved