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 A060900 Number of walks of length n on square lattice, starting at origin, staying on points with x >= 0, y <= x. 8
 1, 2, 7, 21, 78, 260, 988, 3458, 13300, 47880, 185535, 680295, 2649570, 9841260, 38470380, 144263925, 565514586, 2136388436, 8392954570, 31893227366, 125515281892, 479240167224, 1888770070824, 7240285271492, 28569774314536, 109883747363600, 434040802086220 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 A. Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013. A. Bostan, Computer Algebra for Lattice Path Combinatorics, 7th Séminaire Lotharingien de Combinatoire, Ellwangen, March 23-25, 2015. FORMULA The following conjectural formula for this sequence is apparently due to Ira M. Gessel: a(0) = 1, a(2n) = a(2n-1)*(12n+2)/(3n+1), a(2n+1) = a(2n)*(4n+2)/(n+1). G.f.: (hypergeom([ -1/12, 1/4],[2/3],-64*x*(4*x+1)^2/(4*x-1)^4)-1)/(2*x). - Mark van Hoeij, Nov 02 2009 G.f.: (T(x)-1)/(2*x) where T(x) satisfies 27*(4*x-1)^2*T^8 - 18*(4*x-1)^2*T^4 - (128*x^2+192*x+8)*T^2 - (4*x-1)^2 = 0. - Mark van Hoeij, Nov 02 2009 a(n) ~ 4^(n+1) / (sqrt(3) * Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Sep 17 2017 MAPLE b:= proc(n, x, y) option remember;       `if`(x<0 or y>x, 0, `if`(n=0, 1, add(add(        b(n-1, x+i, y+j), j=[-1, 1]), i=[-1, 1])))     end: a:= n-> b(n, 0\$2): seq(a(n), n=0..30);  # Alois P. Heinz, Nov 30 2015 MATHEMATICA (* Conjectural *) a[0]=1; a[n_] := a[n] = If[EvenQ[n], (4*(3*n+1)*a[n-1])/ (3*n+2), (4*n*a[n-1])/(n+1)]; Table[a[n], {n, 0, 26}] (* or, from 1st g.f. *) s = (HypergeometricPFQ[{-1/12, 1/4}, {2/3}, -64*x* (4*x+1)^2/(4*x-1)^4]-1)/(2*x) + O[x]^27; CoefficientList[s, x](* Jean-François Alcover, Nov 30 2015 *) CROSSREFS Cf. A005566, A001700, A060897, A060898, A060899. Sequence in context: A220726 A127540 A319852 * A305850 A151289 A150300 Adjacent sequences:  A060897 A060898 A060899 * A060901 A060902 A060903 KEYWORD nonn AUTHOR David W. Wilson, May 05 2001 EXTENSIONS Entry revised by N. J. A. Sloane at the suggestion of Doron Zeilberger, Sep 13 2007 STATUS approved

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Last modified October 17 19:24 EDT 2019. Contains 328127 sequences. (Running on oeis4.)