OFFSET
0,3
COMMENTS
A Schroeder path of semilength n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Lara Bossinger, Martina Lanini, Following Schubert varieties under Feigin's degeneration of the flag variety, arXiv:1802.04320 [math.RT], 2018.
FORMULA
a(n) = Sum_{k=0..n^2} k * A129179(n,k).
G.f.: (1+z)[1-z-sqrt(1-6z+z^2)]^2/[4z(1-6z+z^2)] (obtained by computing (dG/dt)_{t=1} where G=G(t,z) is defined by G(t,z)=1+zG(t,z)+tzG(t,t^2*z)G(t,z); see A129179).
a(n) = Sum_{k=0..n} (A002315(k)*Sum_{i=0..n-k+1} (binomial(n+1-k,i+2)*binomial(n-k+i,i)))/(n-k+1)). - Vladimir Kruchinin, Mar 02 2016
a(n) ~ 1/2 * (1+sqrt(2))^(2*n+1). - Vaclav Kotesovec, Mar 03 2016
D-finite with recurrence -(n+1)*(2*n-5)*a(n) +3*(4*n+1)*(2*n-5)*a(n-1) +(-76*n^2+228*n-89)*a(n-2) +3*(2*n-1)*(4*n-13)*a(n-3) -(2*n-1)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(2) = 11 because the areas below the Schroeder paths HH, HUD, UDH, UDUD, UHD and UUDD are 0,1,1,2,3 and 4, respectively.
MAPLE
g:=(1+z)*(1-z-sqrt(1-6*z+z^2))^2/4/z/(1-6*z+z^2): gser:=series(g, z=0, 30): seq(coeff(gser, z, n), n=0..24);
MATHEMATICA
CoefficientList[Series[(1 + x)*(1 - x - Sqrt[1 - 6*x + x^2])^2/(4*x*(1 - 6*x + x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 03 2016 *)
PROG
(Maxima)
a(n):=sum((((sqrt(2)+1)^(2*k+1)-(1-sqrt(2))^(2*k)*sqrt(2)+(1-sqrt(2))^(2*k))*sum(binomial(n+1-k, i+2)*binomial(n-k+i, i), i, 0, n-k+1))/(n-k+1), k, 0, n); /* Vladimir Kruchinin, Mar 02 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 08 2007
STATUS
approved