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A005558
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Number of walks on square lattice.
(Formerly M2598)
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6
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1, 1, 3, 6, 20, 50, 175, 490, 1764, 5292, 19404, 60984, 226512, 736164, 2760615, 9202050, 34763300, 118195220, 449141836, 1551580888, 5924217936, 20734762776
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of n-step walks that start at the origin and end on the x-axis, with the constrained to be in the first quadrant. -Benjamin Phillabaum (bphillab(AT)gmail.com) Mar 11 2011
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REFERENCES
| Heinrich Niederhausen, A Note on the Enumeration of Diffusion Walks in the First Octant by Their Number of Contacts with the Diagonal, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
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FORMULA
| a(n) = C(n+1, ceil(n/2))*C(n, floor(n/2)) - C(n+1, ceil((n-1)/2))*C(n, floor((n-1)/2)). - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2004
G.f. (1/(4x^2))*((16*x^2-1)*(hypergeom([1/2, 1/2],[1],16*x^2)+2*x*(4*x-1)*hypergeom([3/2, 3/2],[2],16*x^2))-2*x+1) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Oct 13 2009]
E.g.f. BesselI[1,2*x]*(BesselI[0,2*x]+BesselI[1,2*x])/x. - Benjamin Phillabaum, Feb 25 2011
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PROG
| (PARI) {a(n)=binomial(n+1, ceil(n/2))*binomial(n, floor(n/2)) - binomial(n+1, ceil((n-1)/2))*binomial(n, floor((n-1)/2))}
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CROSSREFS
| See A138350 for a signed version.
Bisections are A000891 and A000888/2.
Cf. A005559-A005562, A093768.
Sequence in context: A052408 A148573 A148574 * A138350 A148575 A148576
Adjacent sequences: A005555 A005556 A005557 * A005559 A005560 A005561
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KEYWORD
| nonn,walk
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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