|
| |
|
|
A026671
|
|
Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).
|
|
21
|
|
|
|
1, 3, 11, 43, 173, 707, 2917, 12111, 50503, 211263, 885831, 3720995, 15652239, 65913927, 277822147, 1171853635, 4945846997, 20884526283, 88224662549, 372827899079, 1576001732485, 6663706588179, 28181895551161, 119208323665543, 504329070986033, 2133944799315027
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
1, 1, 3, 11, 43, 173,... is the unique sequence for which both the Hankel transform of the sequence itself and the Hankel transform of its left shift are the powers of 2 (A000079). For example, det[{{1, 1, 3}, {1, 3, 11}, {3, 11, 43}}] = det[{{1, 3, 11}, {3, 11, 43}, {11, 43, 173}}] = 4. -David Callan,Mar 30 2007
a(n) is the image of F(2n+2) under the Catalan matrix (1,xc(x)) where c(x) is the g.f. ofA000108.
The sequence 1,1,3,... is the image ofA001519under (1,xc(x)). This sequence has g.f. given by 1/(1-x-2x^2/(1-3x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction). (End)
The sequence 1,1,3,... has g.f. 1/(1-x/sqrt(1-4x)), INVERT transform ofA000984.
It is an eigensequence of the sequence array forA000984.(End)
|
|
|
REFERENCES
|
L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: 1/(sqrt(1-4*x)-x).
a(n) = Sum_{i=1..n} a(i-1)*binomial(2*(n-i), n-i) + binomial(2*n, n), n >= 1, a(0)=1. (End)
G.f.: 1/(1 - 3xc(x) + x^2*c(x)^2);
G.f.: 1/(1-3x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction).
a(0) = 1, a(n) = Sum_{k=0..n} (k/(2n-k))*C(2n-k,n-k)*F(2k+2). (End)
G.f.: 1/(1-x/(1-2x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction);
G.f. of 1,1,3,... is 1/(1-x-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)
a(n) = the upper left term in M^n, M = the infinite square production matrix:
3, 2, 0, 0, 0, 0,...
1, 1, 1, 0, 0, 0,...
1, 1, 1, 1, 0, 0,...
1, 1, 1, 1, 1, 0,...
1, 1, 1, 1, 1, 1,...
...
(End)
D-finite with recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 3*(5*n-8)*a(n-2) - 2*(2*n-3)*a(n-3). -Vaclav Kotesovec,Oct 08 2012
|
|
|
MATHEMATICA
|
Table[SeriesCoefficient[1/(Sqrt[1-4*x]-x), {x, 0, n}], {n, 0, 30}] (*Vaclav Kotesovec,Oct 08 2012 *)
|
|
|
PROG
|
(PARI) {a(n)= if(n<0, 0, polcoeff( 1/(sqrt(1 -4*x +x*O(x^n)) -x), n))} /*Michael Somos,Apr 20 2007 */
(PARI) x='x+O('x^66); Vec( 1/(sqrt(1-4*x)-x) ) \\Joerg Arndt,May 04 2013
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(Sqrt(1-4*x)-x) )); //G. C. Greubel,Jul 16 2019
(Sage) (1/(sqrt(1-4*x)-x)).series(x, 30).coefficients(x, sparse=False) #G. C. Greubel,Jul 16 2019
(GAP) a:=[3, 11, 43];; for n in [4..30] do a[n]:=(2*(4*n-3)*a[n-1] - 3*(5*n-8)*a[n-2] - 2*(2*n-3)*a[n-3])/n; od; Concatenation([1], a); #G. C. Greubel,Jul 16 2019
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
