|
| |
|
|
A192446
|
|
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (3,0), (0,1), (0,3).
|
|
1
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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: A319104 A326756 A327927 * A357582 A218940 A005432
Adjacent sequences: A192443 A192444 A192445 * A192447 A192448 A192449
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Joerg Arndt, Jul 01 2011
|
|
|
STATUS
|
approved
|
| |
|
|
