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A026671
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Number of lattice paths from (0,0) to (n,n), n >= 1, with steps (0,1), (1,0) and, when on the diagonal, (1,1).
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10
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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; internal format)
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OFFSET
| 0,2
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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 (callan(AT)stat.wisc.edu), Mar 30 2007
Contribution from Paul Barry (pbarry(AT)wit.ie), 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. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 11 2009]
Contribution from Paul Barry (pbarry(AT)wit.ie), 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)
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REFERENCES
| 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; http://repository.wit.ie/1693/1/AoifeThesis.pdf
L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
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LINKS
| Jean-Christophe Aval, Adrien Boussicault and Sandrine Dasse-Hartaut, The tree structure in staircase tableaux, Arxiv preprint arXiv:1109.4907, 2011
Miklos Bona, The permutation classes equinumerous to the smooth class, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
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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 (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 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
Contribution from Paul Barry (pbarry(AT)wit.ie), 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). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 11 2009]
Contribution from Paul Barry (pbarry(AT)wit.ie), 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
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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 */
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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.
Sequence in context: A140803 A084643 A007583 * A026876 A151090 A059278
Adjacent sequences: A026668 A026669 A026670 * A026672 A026673 A026674
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu); Miklos Bona (bona(AT)math.ufl.edu)
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