A175962 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|k>0,r>0} which never go above the line y=x.

1, 2, 10, 63, 454, 3539, 29008, 246255, 2145722, 19078536, 172402396, 1578687082, 14616730080, 136606848093, 1287022395324, 12210382758519, 116553763025178, 1118580919711060, 10786838228669692, 104469304517331666, 1015700422725526916

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Joseph P. S. Kung and Anna de Mier, Rook and queen paths with boundaries, arXiv:1109.1806 [math.CO], 2011.

J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From N. J. A. Sloane, Dec 27 2012

Formula

G.f.: ((1-t)*(1+t-4*t^2)-(1-t)^2*sqrt(1-12*t+16*t^2))/(2*t*(2 - 3*t)^2). [Kung-de Mier]. - corrected by Vaclav Kotesovec, Sep 07 2012

Apparently 2*n*(n+1)*a(n) -n*(29*n-10)*a(n-1) +19*n*(5*n-7)*a(n-2) -2*n*(58*n-149)*a(n-3) +48*n*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 24 2012

a(n) ~ 5/2*sqrt(246*sqrt(5)-550)/sqrt(Pi) * (6+2*sqrt(5))^n/n^(3/2). - Vaclav Kotesovec, Nov 01 2012

Equivalently, a(n) ~ 5^(5/4) * 2^(2*n) * phi^(2*n - 5) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

Mathematica

Table[SeriesCoefficient[((1-t)*(1+t-4t^2)-(1-t)^2*Sqrt[1-12t+16t^2])/(2t*(2-3t)^2), {t, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2012 *)

Prog

(PARI) x='x+O('x^50); Vec(((1-t)*(1+t-4*t^2)-(1-t)^2*sqrt(1-12*t+16*t^2))/(2*t*(2 - 3*t)^2)) \\ G. C. Greubel, Mar 22 2017

Crossrefs

Cf. A175912, A175939.

Sequence in context: A361829 A361494 A371546 * A183165 A129130 A245519

Adjacent sequences: A175959 A175960 A175961 * A175963 A175964 A175965

Keyword

nonn

Author

Eric Werley, Dec 06 2010

Extensions

Edited by N. J. A. Sloane, Sep 24 2011

Minor edits by Vaclav Kotesovec, Mar 31 2014

Status

approved