|
Search: id:A108442
|
|
|
| A108442 |
|
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 only u steps among the steps leading to the first d step. |
|
+0
3
|
|
| 1, 1, 3, 15, 97, 721, 5827, 49759, 441729, 4035937, 37702723, 358474735, 3457592161, 33748593841, 332730216579, 3308635650495, 33145196426753, 334193815799233, 3388807714823043, 34537227997917391, 353578650475659617
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
REFERENCES
|
Problem 10658, American Math. Monthly, 107, 2000, 368-370.
|
|
FORMULA
|
G.f.=1/(1-zA), 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)=3 because we have udud, udUdd, and uudd.
|
|
MAPLE
|
A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: gser:=series(1/(1-z*A), z=0, 30): 1, seq(coeff(gser, z^n), n=1..25);
|
|
CROSSREFS
|
Column 0 of A108441.
Cf. A027307, A108441.
Sequence in context: A112913 A109283 A079689 this_sequence A060148 A132437 A128081
Adjacent sequences: A108439 A108440 A108441 this_sequence A108443 A108444 A108445
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 08 2005
|
|
|
Search completed in 0.002 seconds
|
