A175939 Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|k>0,0<r<=2} which never go above the line y=x.

1, 2, 10, 62, 448, 3495, 28640, 242946, 2114829, 18783658, 169546150, 1550728135, 14340859992, 133867779775, 1259689173181, 11936488052113, 113799596287017, 1090803942244627, 10505978544362607, 101623141479156708, 986801698075230291

Offset

0,2

References

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

Links

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

Formula

a(n) ~ b*c^n/n^(3/2), where c = 10.458904071481665... is the root of the equation x^4-10*x^3-5*x^2+2*x+1=0 and b = sqrt(2*(1-5*c-15*c^2+2*c^3) /c^3)*(-5 - 4*c + 21*c^2 + 27*c^3) / (44*c^3*sqrt(Pi)) = 0.3791408579... - Vaclav Kotesovec, Aug 10 2013

G.f.: ((x^4+2*x^3-5*x^2-10*x+1)^(1/2)-x^2-3*x-1)/(2*x*(x-1)*(x+2)^2). - Mark van Hoeij, Apr 16 2013

Mathematica

Flatten[{1, RecurrenceTable[{(n+2)*a[n]+(3*n+7)*a[n+1]-(5*n+24)*a[n+2]-(19*n+90)*a[n+3]+(n+3)*a[n+4]+(21*n+116)*a[n+5]-2*(n+7)*a[n+6]==0, a[1]==2, a[2]==10, a[3]==62, a[4]==448, a[5]==3495, a[6]==28640}, a, {n, 20}]}] (* Vaclav Kotesovec, Sep 07 2012 *)

Crossrefs

Cf. A175934, A175935, A175936, A175937.

Sequence in context: A107841 A175936 A175937 * A192942 A307364 A141140

Adjacent sequences: A175936 A175937 A175938 * A175940 A175941 A175942

Keyword

nonn

Author

Eric Werley, Dec 06 2010

Extensions

Minor edits Vaclav Kotesovec, Mar 31 2014

Status

approved