A129180 Total area below all Schroeder paths of semilength n.

0, 1, 11, 85, 583, 3785, 23843, 147437, 900559, 5453457, 32816315, 196531781, 1172634391, 6976059865, 41401814099, 245230349021, 1450162049695, 8563622372129, 50510963880299, 297627067200821, 1752169739791591, 10307304302433513, 60592569330907523

Offset: 0

Author

Emeric Deutsch, Apr 08 2007

Keywords

• nonn

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.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Lara Bossinger and Martina Lanini, Following Schubert varieties under Feigin's degeneration of the flag variety, arXiv:1802.04320 [math.RT], 2018.

Crossrefs

Cf. A002315, A006319, A129179.

Programs

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 *)

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 */

Formula

a(n) = Sum_{k=0..n^2} k * A129179(n,k).

G.f.: (1+z)*(1-z-sqrt(1-6*z+z^2))^2/(4*z*(1-6*z+z^2)) (obtained by computing (dG/dt)_{t=1} where G=G(t,z) is defined by G(t,z) = 1+z*G(t,z)+t*z*G(t,t^2*z)G(t,z); see A129179).

a(n) = Sum_{k=0..n} 2*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