A192364 Number of lattice paths from (0,0) to (n,n) using steps (0,1),(0,2),(1,0),(2,0),(1,1).

1, 3, 21, 157, 1239, 10047, 82951, 693603, 5854581, 49778997, 425712429, 3657968097, 31555053921, 273109567797, 2370474720369, 20625186298269, 179841473895447, 1571088267426447, 13747953837604959, 120482775658910763, 1057293764707074027, 9289536349244758791, 81709329486947791419

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

From Vaclav Kotesovec, Oct 24 2012: (Start)

G.f.: (3 - 6*x + sqrt(-1 + 4*x*(9*x-11) + 4*sqrt(1-x)*sqrt(5+4*x)*sqrt(9*x-1))) / (sqrt(10+8*x)*sqrt((1-x)*(1-9*x))*(4*x*(9*x-11)-1+4*sqrt(1-x)*sqrt(5+4*x)*sqrt(9*x-1))^(1/4))

D-finite with recurrence: 15*(n-1)*n*a(n) = (n-1)*(133*n-54)*a(n-1) + (31*n^2 - 177*n + 224)*a(n-2) - (113*n^2 - 295*n + 144)*a(n-3) - 18*(n-3)*(2*n-5)*a(n-4)

a(n) ~ 3^(2*n+3/2)/(2*sqrt(14*Pi*n))

(End)

a(n) = A091533(2*n,n) for n >= 0. - Paul D. Hanna, Dec 11 2018

a(n) = [x^n*y^n] 1/(1 - x - y - x^2 - x*y - y^2) for n >= 0. - Paul D. Hanna, Dec 11 2018

Mathematica

FullSimplify[CoefficientList[Series[(3-6*x+Sqrt[-1+4*x*(9*x-11)+4*Sqrt[1-x]*Sqrt[5+4*x]*Sqrt[9*x-1]])/(Sqrt[10+8*x]*Sqrt[(1-x)*(1-9*x)]*(4*x*(9*x-11)-1+4*Sqrt[1-x]*Sqrt[5+4*x]*Sqrt[9*x-1])^(1/4)), {x, 0, 10}], x]]

Prog

(PARI) /* same as in A092566 but use */
steps=[[0, 1], [0, 2], [1, 0], [2, 0], [1, 1]];
/* Joerg Arndt, Jun 30 2011 */

Crossrefs

Cf. A091533.

Sequence in context: A226560 A026333 A205773 * A286918 A358953 A189508

Adjacent sequences: A192361 A192362 A192363 * A192365 A192366 A192367

Keyword

nonn,walk

Author

Eric Werley, Jun 29 2011

Extensions

Terms > 425712429 by Joerg Arndt, Jun 30 2011

Status

approved