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

0,2

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)

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A326756 A362696 A327927 * A357582 A218940 A005432

Adjacent sequences: A192443 A192444 A192445 * A192447 A192448 A192449

Keyword

nonn

Author

Joerg Arndt, Jul 01 2011

Status

approved