A134424 Area under all paths in the first quadrant from (0,0) to (n,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0).

0, 0, 1, 4, 21, 80, 316, 1152, 4186, 14812, 52020, 180616, 623338, 2138040, 7302035, 24842736, 84262609, 285052676, 962184359, 3241616628, 10903119167, 36619715860, 122837641530, 411588875136, 1377735161776, 4607695277512

Offset: 0

Author

Emeric Deutsch, Oct 25 2007

Keywords

• nonn

Examples

			a(3)=4 because the areas under the paths hhh, hH, Hh, hUD, UhD and UDh are 0,0,0,1,2 and 1, respectively.
		

Crossrefs

Cf. A128720, A134423.

Programs

Maple

g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/z^2: G:=z^2*(1+z^2)*g^2/((1+z-z^2)*(1-3*z-z^2)): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..25);

Formula

a(n) = Sum_{k>=0} k * A134423(n,k).

G.f.: z^2*(1+z^2)*g^2/((1+z-z^2)*(1-3*z-z^2)), where g=1+z*g+z^2*g+z^2*g^2 (g is the g.f. of A128720).

Conjecture D-finite with recurrence -(n+2)*(5*n-7)*a(n) -(n+1)*(5*n-127)*a(n-1) +(135*n^2-655*n-42)*a(n-2) +2*(5*n^2-275*n-108)*a(n-3) +(-725*n^2+4941*n-5734)*a(n-4) +(-235*n^2+1880*n-1173)*a(n-5) +(725*n^2-6659*n+12606)*a(n-6) +2*(5*n^2+195*n-1988)*a(n-7) +(-135*n^2+1505*n-3358)*a(n-8) -(5*n+87)*(n-9)*a(n-9) +(5*n-33)*(n-10)*a(n-10)=0. - R. J. Mathar, Jul 24 2022