A192446 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (3,0), (0,1), (0,3).

1, 2, 6, 30, 154, 768, 3906, 20232, 105750, 556328, 2943432, 15646932, 83500126, 447057380, 2400249624, 12918250836, 69674241654, 376489511460, 2037768450480, 11045915485740, 59955446568276, 325821729044784, 1772588671356204, 9653187691115640, 52617711157401186, 287051310425050668

Offset: 0

Author

Joerg Arndt, Jul 01 2011

Keywords

• nonn

Comments

Diagonal of rational function 1/(1 - (x + y + x^3 + y^3)). - Gheorghe Coserea, Aug 06 2018

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1344 (first 304 terms from Gheorghe Coserea)

Crossrefs

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A192368.

Programs

Maple

REL := 3*s^2*x^2-(1-2*s^2+2*s^3)*x-s*(s^3+2*s^2+s-1);
ogf := sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3);
series(eval(ogf, s=RootOf(REL,s)),x=0,30);  # Mark van Hoeij, Apr 17 2013
# second Maple program:
b:= proc(x, y) option remember; `if`(y=0, 1, add((p->
      `if`(p[1]<0, 0, b(p[1], p[2])))(sort([x, y]-h)),
        h=[[1, 0], [0, 1], [3, 0], [0, 3]]))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 28 2018

Mathematica

a[0, 0] = 1; a[n_, k_] /; n >= 0 && k >= 0 := a[n, k] = a[n, k-1] + a[n, k-3] + a[n-1, k] + a[n-3, k]; a[, ] = 0;
a[n_] := a[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Oct 06 2019 *)

PARI

/* same as in A092566 but use */
steps=[[1,0], [3,0], [0,1], [0,3]];
/* Joerg Arndt, Jun 30 2011 */

PARI

seq(N) = {
  my(x='x + O('x^N), d=16*x^6 + 16*x^5 + 16*x^4 - 8*x^3 - 4*x^2 + 1,
     s=serreverse((1 - 2*x^2 + 2*x^3 - sqrt(d))/(6*x^2)));
  Vec(sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3));
};
seq(26) \\ Gheorghe Coserea, Aug 06 2018

Formula

G.f.: sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3) where s is a function satisfying 3*s^2*x^2-(1-2*s^2+2*s^3)*x-s*(s^3+2*s^2+s-1)=0. - Mark van Hoeij, Apr 17 2013

From Gheorghe Coserea, Aug 06 2018: (Start)

G.f. y=A(x) satisfies:

0 = (4*x^3 + 8*x^2 + 4*x - 1)^4*(108*x^3 - 108*x^2 + 36*x - 31)^2*y^8 + 4*(4*x^3 + 8*x^2 + 4*x - 1)^3*(36*x^3 + 36*x^2 - 4*x - 13)*(108*x^3 - 108*x^2 + 36*x - 31)*y^6 + 2*(4*x^3 + 8*x^2 + 4*x - 1)^2*(2160*x^6 + 4320*x^5 + 1872*x^4 - 1784*x^3 - 1576*x^2 + 472*x + 431)*y^4 + 4*(4*x^3 + 8*x^2 + 4*x - 1)*(112*x^6 + 448*x^5 + 688*x^4 + 456*x^3 + 96*x^2 + 40*x + 55)*y^2 + (4*x^3 + 12*x^2 + 12*x + 3)^2.

0 = (4*x^3 + 8*x^2 + 4*x - 1)*(108*x^3 - 108*x^2 + 36*x - 31)*(270*x^4 + 180*x^3 + 144*x^2 - 225*x - 59)*y''' + (1283040*x^9 + 1924560*x^8 + 1080864*x^7 - 1425816*x^6 - 2135376*x^5 + 33048*x^4 + 702468*x^3 + 134520*x^2 + 43892*x + 30575)*y'' + 30*(111780*x^8 + 149040*x^7 + 120960*x^6 - 122094*x^5 - 172206*x^4 - 6012*x^3 + 36615*x^2 + 10298*x - 1541)*y' + 60*(29160*x^7 + 34020*x^6 + 36288*x^5 - 43092*x^4 - 45882*x^3 - 6462*x^2 + 1890*x + 913)*y.

(End)