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 A125190 Number of ascents in all Schroeder paths of length 2n. 1
 0, 1, 6, 32, 170, 912, 4942, 27008, 148626, 822560, 4573910, 25534368, 143027898, 803467056, 4524812190, 25537728000, 144411206178, 818017823808, 4640757865126, 26364054632480, 149959897539018, 853941394691792, 4867745532495086, 27773897706129792 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A Schroeder path of length 2n is a lattice path in the first quadrant, from the origin to the point (2n, 0) and consisting of steps U = (1, 1), D = (1, -1) and H = (2, 0); an ascent in a Schroeder path is a maximal strings of U steps. a(n) is the number of points at L1 distance n - 2 from any point in Z^n, for n >= 2. - Shel Kaphan, Mar 24 2023 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..400 Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374. FORMULA a(n) = Sum_{k=0..n} k * A090981(n, k). G.f.: z*R*(1 + z*R)/sqrt(1 - 6*z + z^2), where R = 1 + z*R + z*R^2, i.e., R = (1 - z -sqrt(1 - 6*z + z^2))/(2*z). D-finite Recurrence: 2*n*(17*n - 26)*a(n) = 3*(68*n^2 - 137*n + 66)*a(n-1) - 2*(17*n^2 - 34*n - 48)*a(n-2) + 3*(n - 4)*a(n-3). - Vaclav Kotesovec, Oct 19 2012 a(n) ~ 2^(-3/4)*(3 + 2*sqrt(2))^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 19 2012 a(n) = Sum_{i=0..n-1} binomial(n+1, n-i-1) * binomial(n+i, n)). - Vladimir Kruchinin, Feb 05 2013 a(n) = (n*(n+1)/2)*hypergeometric([1-n, n+1], [3], -1). - Peter Luschny, Sep 17 2014 a(n) = A026002(n) - A190666(n-2) for n >= 2. - Shel Kaphan, Mar 24 2023 a(n) = ((n+1)/2) * A006319(n-1). - Vladimir Kruchinin, Apr 27 2024 EXAMPLE a(2) = 6 because the Schroeder paths of length 4 are HH, H(U)D, (U)DH, (U)D(U)D, (U)HD and (UU)DD, having a total of 6 ascents (shown between parentheses). MAPLE R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*R*(1+z*R)/sqrt(1-6*z+z^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..25); # second Maple program: a:= proc(n) option remember; `if`(n<3, [0, 1, 6][n+1], ((204*n^2-411*n+198)*a(n-1) +(-34*n^2+68*n+96)*a(n-2) +(3*n-12)*a(n-3))/(2*n*(17*n-26))) end: seq(a(n), n=0..30); # Alois P. Heinz, Oct 20 2012 MATHEMATICA CoefficientList[Series[x*(1-x-Sqrt[1-6*x+x^2])/(2*x)*(1+x*(1-x-Sqrt[1-6*x+x^2])/(2*x))/Sqrt[1-6*x+x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *) a[n_] := Sum[ Binomial[n+1, n-i-1]*Binomial[n+i, n], {i, 0, n-1}]; (* or *) a[n_] := Hypergeometric2F1[1-n, 1+n, 3, -1]*n*(n+1)/2; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 05 2013, after Vladimir Kruchinin *) PROG (Sage) A125190 = lambda n : (n^2+n)*hypergeometric([1-n, n+1], [3], -1)/2 [round(A125190(n).n(100)) for n in (0..23)] # Peter Luschny, Sep 17 2014 CROSSREFS Cf. A006319, A090981, A008288, A026002, A190666. -2-diagonal of A266213 for n>=1. Sequence in context: A199699 A306900 A137637 * A264460 A180037 A277742 Adjacent sequences: A125187 A125188 A125189 * A125191 A125192 A125193 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 20 2006 STATUS approved

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Last modified August 12 15:11 EDT 2024. Contains 375113 sequences. (Running on oeis4.)