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A125190 Number of ascents in all Schroeder paths of length 2n.

%I A125190 #52 Sep 17 2024 21:07:20
%S A125190 0,1,6,32,170,912,4942,27008,148626,822560,4573910,25534368,143027898,
%T A125190 803467056,4524812190,25537728000,144411206178,818017823808,
%U A125190 4640757865126,26364054632480,149959897539018,853941394691792,4867745532495086,27773897706129792
%N A125190 Number of ascents in all Schroeder paths of length 2n.
%C A125190 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.
%C A125190 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
%H A125190 Alois P. Heinz, <a href="/A125190/b125190.txt">Table of n, a(n) for n = 0..400</a>
%H A125190 Emanuele Munarini, <a href="https://www.emis.de/journals/INTEGERS/papers/j29/j29.Abstract.html">Combinatorial properties of the antichains of a garland</a>, Integers, 9 (2009), 353-374.
%F A125190 a(n) = Sum_{k=0..n} k * A090981(n, k).
%F A125190 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).
%F A125190 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
%F A125190 a(n) ~ 2^(-3/4)*(3 + 2*sqrt(2))^n/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 19 2012
%F A125190 a(n) = Sum_{i=0..n-1} binomial(n+1, n-i-1) * binomial(n+i, n). - _Vladimir Kruchinin_, Feb 05 2013
%F A125190 a(n) = (n*(n+1)/2)*hypergeometric([1-n, n+1], [3], -1). - _Peter Luschny_, Sep 17 2014
%F A125190 a(n) = A026002(n) - A190666(n-2) for n >= 2. - _Shel Kaphan_, Mar 24 2023
%F A125190 a(n) = ((n+1)/2) * A006319(n-1). - _Vladimir Kruchinin_, Apr 27 2024
%e A125190 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).
%p A125190 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);
%p A125190 # second Maple program:
%p A125190 a:= proc(n) option remember;
%p A125190       `if`(n<3, [0,1,6][n+1], ((204*n^2-411*n+198)*a(n-1)
%p A125190        +(-34*n^2+68*n+96)*a(n-2) +(3*n-12)*a(n-3))/(2*n*(17*n-26)))
%p A125190     end:
%p A125190 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 20 2012
%t A125190 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 *)
%t A125190 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_ *)
%o A125190 (Sage)
%o A125190 A125190 = lambda n : (n^2+n)*hypergeometric([1-n, n+1], [3], -1)/2
%o A125190 [round(A125190(n).n(100)) for n in (0..23)] # _Peter Luschny_, Sep 17 2014
%Y A125190 Cf. A006319, A090981, A008288, A026002, A190666.
%Y A125190 -2-diagonal of A266213 for n>=1.
%K A125190 nonn
%O A125190 0,3
%A A125190 _Emeric Deutsch_, Dec 20 2006