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Search: id:A026671
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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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+0
7
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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; listen)
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OFFSET
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0,2
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COMMENT
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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
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LINKS
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J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Miklos Bona, The permutation classes equinumerous to the smooth class, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.
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FORMULA
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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
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PROGRAM
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(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
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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: A034477 A084643 A007583 this_sequence A026876 A059278 A103821
Adjacent sequences: A026668 A026669 A026670 this_sequence A026672 A026673 A026674
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu); Miklos Bona (bona(AT)math.ufl.edu)
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