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A110190

Number of (1,0)-steps on the lines y=0 and y=1 in all Schroeder paths of length 2n (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis).

0, 1, 5, 24, 116, 568, 2820, 14184, 72180, 371112, 1925380, 10068728, 53023860, 280969560, 1497072132, 8016213960, 43114424308, 232817773640, 1261793848836, 6861179441880, 37421756333172, 204671007577464, 1122275850740996, 6168352091629864, 33977333521770996, 187539324760522728

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OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

FORMULA

a(n) = Sum_{k=0..n} k*A110189(n,k).

G.f.: x*(1-x-2*x*R+x^2+2*x^2*R+x^2*R^2)/(1-3*x-x*R+x^2+x^2*R)^2, where R = 1+x*R+x*R^2 = (1-x-sqrt(1-6*x+x^2))/(2*x) is the g.f. for the large Schroeder numbers (A006318).

Recurrence: (n+2)*(n+3)*a(n) = (5*n^2+29*n+10)*a(n-1) + (5*n^2-59*n+142)*a(n-2) - (n-6)*(n-5)*a(n-3). - Vaclav Kotesovec, Oct 18 2012

a(n) ~ 3*2^(1/4)*(3+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

G.f. A(x) satisfies x^2*A(x)^2 = (x^4 - 7*x^3 + 12*x^2 - 7*x + 1)*A(x) + (-x^3 + 2*x^2 - x). - Joerg Arndt, May 16 2013

a(n) = Sum_{k=0..n} ((k+1)*Sum_{i=0..n-k} (binomial(n+1,n-k-i)*binomial(n+i,n))/ (n+1)*A113127(k)). - Vladimir Kruchinin, Mar 13 2016

EXAMPLE

a(2)=5 because in the 6 (=A006318(2)) Schroeder paths of length 4, namely, HH, HUD, UDH, UDUD, UHD, UUDD, all 5 H-steps are at levels 0 or 1.

MAPLE

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-2*z*R+z^2+2*z^2*R+z^2*R^2)/(1-3*z-z*R+z^2+z^2*R)^2: Gser:=series(G, z=0, 30): 0, seq(coeff(Gser, z^n), n=1..26);

MATHEMATICA

CoefficientList[Series[x*(1-x-2*x*((1-x-Sqrt[1-6*x+x^2])/(2*x))+x^2+2*x^2*((1-x-Sqrt[1-6*x+x^2])/(2*x))+x^2*((1-x-Sqrt[1-6*x+x^2])/(2*x))^2)/(1-3*x-x*((1-x-Sqrt[1-6*x+x^2])/(2*x))+x^2+x^2*((1-x-Sqrt[1-6*x+x^2])/(2*x)))^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

PROG

(PARI)

x = 'x+O('x^66);

R = (1-x-sqrt(1-6*x+x^2))/(2*x);

gf = x*(1-x-2*x*R+x^2+2*x^2*R+x^2*R^2)/(1-3*x-x*R+x^2+x^2*R)^2;

concat([0], Vec(gf))

\\ Joerg Arndt, May 16 2013

(Maxima)

a113127(n):=if n=0 then 1 else if n=1 then 3 else 4*n-2;

a(n):=sum((k+1)*sum(binomial(n+1, n-k-i)*binomial(n+i, n), i, 0, n-k)/(n+1)*a113127(k), k, 0, n); /* Vladimir Kruchinin, Mar 13 2016 */

CROSSREFS

Cf. A006318, A110189, A113127.