|
Search: id:A108767
|
|
|
| A108767 |
|
Triangle read by rows: T(n,k) is 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=(1,1), d=(1,-2) and have k peaks (i.e. ud's). |
|
+0
2
|
|
| 1, 1, 2, 1, 6, 5, 1, 12, 28, 14, 1, 20, 90, 120, 42, 1, 30, 220, 550, 495, 132, 1, 42, 455, 1820, 3003, 2002, 429, 1, 56, 840, 4900, 12740, 15288, 8008, 1430, 1, 72, 1428, 11424, 42840, 79968, 74256, 31824, 4862, 1, 90, 2280, 23940, 122094, 325584, 465120
(list; table; graph; listen)
|
|
|
OFFSET
|
1,3
|
|
|
COMMENT
|
Row sums yield A001764. T(n,n)=A000108(n) (the Catalan numbers). sum(kT(n,k),k=1..n)=A025174(n).
|
|
FORMULA
|
G.f.=T-1, where T=T(t, z) satisfies T=1+zT^2*(T-1+t).
|
|
EXAMPLE
|
T(3,2)=6 because we have uuduuuudd, uuuduuudd, uuuuduudd, uuuudduud, uuuuududd, and uuuuuddud.
Triangle starts:
1;
1,2;
1,6,5;
1,12,28,14;
|
|
MAPLE
|
T:=(n, k)->binomial(n, k)*binomial(2*n, k-1)/n: for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
|
|
CROSSREFS
|
Cf. A001764, A000108, A025174.
Sequence in context: A128728 A084950 A066654 this_sequence A046817 A008970 A055896
Adjacent sequences: A108764 A108765 A108766 this_sequence A108768 A108769 A108770
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 24 2005
|
|
|
Search completed in 0.002 seconds
|
