|
| |
|
|
A108448
|
|
Number of peaks of the form ud in all 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).
|
|
1
| |
|
|
1, 7, 61, 575, 5641, 56695, 579125, 5984767, 62390545, 654862247, 6911195501, 73265596607, 779594526361, 8321683861015, 89070157349221, 955598531432447, 10273391096237089, 110647714508386375, 1193641560393864605
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| a(n)=sum(k*A108446(n,k),k=1..n). Example: a(3)=1*32+2*13+3*1=61.
|
|
|
REFERENCES
| Problem 10658, American Math. Monthly, 107, 2000, 368-370.
|
|
|
FORMULA
| G.f.=zA/(1-2zA-3zA^2), where A=1+zA^2+zA^3 or, equivalently, A=(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)=7 because in the ten paths (ud)(ud), (ud)Udd, u(ud)d, uUddd, Udd(ud), UddUdd, Ud(ud)d, UdUddd, U(ud)dd and UUdddd (see A027307) we have 7 ud's (shown between parentheses).
|
|
|
MAPLE
| A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=z*A/(1-2*z*A-3*z*A^2): Gser:=series(G, z=0, 25): seq(coeff(Gser, z^n), n=1..23);
|
|
|
CROSSREFS
| Cf. A027307, A108446, A108426, A108427.
Sequence in context: A104093 A015572 A066443 * A098659 A199686 A113718
Adjacent sequences: A108445 A108446 A108447 * A108449 A108450 A108451
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 10 2005
|
| |
|
|
