|
| |
|
|
A108452
|
|
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 of the first kind (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).
|
|
1
| |
|
|
1, 1, 6, 44, 344, 2856, 24816, 223016, 2056256, 19344472, 184956240, 1792088296, 17558218048, 173659691928, 1731556718224, 17387182158184, 175670235597120, 1784561125349464, 18216639085961552, 186762117058304104
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Also 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 of the second kind (a pyramid of the second kind is a sequence U^pd^(2p) for some positive integer p, starting at the x-axis). Column 0 of A108451.
|
|
|
REFERENCES
| Problem 10658, American Math. Monthly, 107, 2000, 368-370.
|
|
|
FORMULA
| G.f.=(1-z)/[1-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).
|
|
|
EXAMPLE
| a(2)=6 because the paths uUddd, UddUdd, Ududd, UdUddd, Uuddd and UUdddd have no pyramids of the first kind.
|
|
|
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*(1-z)*A*(1+A)): gser:=series(g, z=0, 24): 1, seq(coeff(gser, z^n), n=1..21);
|
|
|
CROSSREFS
| Cf. A027307, A108451, A108449, A108445.
Sequence in context: A114935 A115969 A082412 * A005591 A052204 A147688
Adjacent sequences: A108449 A108450 A108451 * A108453 A108454 A108455
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 11 2005
|
| |
|
|
