A108449 Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and having no pyramids (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).

1, 0, 4, 32, 252, 2112, 18484, 166976, 1545548, 14583808, 139774180, 1356966240, 13316740764, 131890671680, 1316627340564, 13234192747648, 133829733962732, 1360586260341248, 13898403178004420, 142578916276009632

Offset: 0

Author

Emeric Deutsch, Jun 11 2005

Keywords

• nonn

Comments

Column 0 of A108445.

Examples

			a(2)=4 because the paths uUddd, Ududd, UdUddd and Uuddd have no pyramids.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..100
• Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Crossrefs

Cf. A027307, A108445.

Programs

Maple

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=(1-z)/(1+z-z*(1-z)*A*(1+A)): gser:=series(g,z=0,24): 1,seq(coeff(gser,z^n),n=1..21);

PARI

{a(n)=local(y=1); for(i=1, n, y = -(-1 + 3*x - 3*x^2 + x^3 + 3*x*y - 9*x^2*y + 5*x^3*y - 5*x*y^2 - x^2*y^2 + 5*x^3*y^2 + x^4*y^2 - x*y^3 + 9*x^2*y^3 - 3*x^3*y^3 + 3*x^4*y^3) + (O(x^n))^4); polcoeff(y, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 18 2014

Formula

G.f.=(1-z)/[1+z-z(1-z)A(1+A)], where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).

G.f. y(x) satisfies: -1 + 3*x - 3*x^2 + x^3 + y + 3*x*y - 9*x^2*y + 5*x^3*y - 5*x*y^2 - x^2*y^2 + 5*x^3*y^2 + x^4*y^2 - x*y^3 + 9*x^2*y^3 - 3*x^3*y^3 + 3*x^4*y^3 = 0. - Vaclav Kotesovec, Mar 18 2014

a(n) ~ (11+5*sqrt(5))^n * sqrt(1738885 + 811683*sqrt(5)) / (961*sqrt(5*Pi) *n^(3/2)*2^(n+3/2)). - Vaclav Kotesovec, Mar 18 2014

Conjecture D-finite with recurrence +n*(2*n+1)*(431*n-2895)*a(n) +2*(-9395*n^3+68622*n^2-64084*n+26109)*a(n-1) +2*(59288*n^3-508196*n^2+1044822*n-574587)*a(n-2) +2*(-94965*n^3+1070605*n^2-3607435*n+3485484)*a(n-3) +4*(29036*n^3-351474*n^2+1402336*n-1970505)*a(n-4) +6*(-6703*n^3+63052*n^2-99178*n-237177)*a(n-5) +6*(1012*n^3-14914*n^2+74580*n-127341)*a(n-6) +6*(1127*n^3-21429*n^2+135199*n-282762

)*a(n-7) +9*(29*n-165)*(2*n-15)*(n-8)*a(n-8)=0. - R. J. Mathar, Jul 26 2022