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A026671 Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1). 20
1, 3, 11, 43, 173, 707, 2917, 12111, 50503, 211263, 885831, 3720995, 15652239, 65913927, 277822147, 1171853635, 4945846997, 20884526283, 88224662549, 372827899079, 1576001732485, 6663706588179, 28181895551161, 119208323665543 (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

From Paul Barry, Jan 25 2009: (Start)

a(n) is the image of F(2n+2) under the Catalan matrix (1,xc(x)) where c(x) is the g.f. of A000108.

The sequence 1,1,3,... is the image of A001519 under (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)

Binomial transform of A111961. [Philippe Deléham, Feb 11 2009]

From Paul Barry, Nov 03 2010: (Start)

The sequence 1,1,3,... has g.f. 1/(1-x/sqrt(1-4x)), INVERT transform of A000984.

It is an eigensequence of the sequence array for A000984. (End)

REFERENCES

L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Jean-Christophe Aval, Adrien Boussicault and Sandrine Dasse-Hartaut, The tree structure in staircase tableaux, arXiv:1109.4907 [math.CO], 2011-2013.

Miklos Bona, The permutation classes equinumerous to the smooth class, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.

David Callan, Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

FORMULA

G.f.: 1/(sqrt(1-4*x)-x); a(n)= sum(a(i-1)*binomial(2*(n-i), n-i), i=1..n) + binomial(2*n, n), n >= 1, a(0)=1. - Wolfdieter Lang, Mar 21 2000

G.f.: 1/(1 -x -2*x*c(x)) where c(x) = g.f. for Catalan numbers A000108. - Michael Somos, Apr 20 2007

From Paul Barry, Jan 25 2009: (Start)

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)

a(n) = Sum_{k, 0<=k<=n} A039599(n,k)*A000045(k+2). [Philippe Deléham, Feb 11 2009]

From Paul Barry, Feb 08 2009: (Start)

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,...

...

- Gary W. Adamson, Jul 14 2011

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

a(n) ~ (2+sqrt(5))^n/sqrt(5). - Vaclav Kotesovec, Oct 08 2012

MATHEMATICA

Table[SeriesCoefficient[1/(Sqrt[1-4*x]-x), {x, 0, n}], {n, 0, 20}] (* 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

CROSSREFS

a(n)=T(2n-1,n-1), T given by A026736, a(n)=T(2n,n), T given by A026670, a(n)=T(2n+1,n+1), T given by A026725. Row sums of triangle A054335.

Cf. A026781.

Sequence in context: A246758 A084643 A007583 * A026876 A270447 A151090

Adjacent sequences:  A026668 A026669 A026670 * A026672 A026673 A026674

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling; Miklos Bona (bona(AT)math.ufl.edu)

STATUS

approved

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Last modified August 21 00:36 EDT 2017. Contains 290855 sequences.