|
Search: id:A114848
|
|
|
| A114848 |
|
Triangle read by rows T(n,k) = the number of Dyck paths of semilength n with k UUDDU's, 0<=k<=[(n-1)/2]. |
|
+0
1
|
|
| 1, 1, 2, 4, 1, 10, 4, 28, 13, 1, 82, 44, 6, 248, 153, 27, 1, 770, 536, 116, 8, 2440, 1889, 486, 46, 1, 7858, 6696, 1992, 240, 10, 25644, 23849, 8042, 1180, 70, 1, 84618, 85276, 32124, 5552, 430, 12, 281844, 305933, 127287, 25306, 2430, 99, 1, 946338, 1100692
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
Row sums are Catalan numbers A000108.
|
|
FORMULA
|
T(n,k) = Sum((-1)^j * binomial(n-1-(j+k), j+k) * binomial(j + k, k) * A000108(n-2(j+k)), j=0..[(n-1)/2]-k). G.f. G = G(t,z) satisfies G = C(z/(z^2(1-t)+1)), where C(z) is g.f. of Catalan numbers.
|
|
EXAMPLE
|
T(4,1) = 4 because there exist 4 Dyck paths with one occurrence of UUDDU : UDUUDDUD, UUDDUDUD, UUDDUUDD, UUUDDUDD.
|
|
MATHEMATICA
|
For[n = 1, n <= 20, n++, For[k = 0, k <= Floor[(n - 1)/2], k++, Print[Sum[(-1)^j * Binomial[n - 1 - (j + k), j + k] * Binomial[j + k, k] * Binomial[2(n - 2(j + k)), n - 2(j + k)]/(n - 2(j + k) + 1), {j, 0, Floor[(n - 1)/2] - k}]]]]
|
|
CROSSREFS
|
Sequence in context: A135306 A102405 A114506 this_sequence A135330 A135328 A048941
Adjacent sequences: A114845 A114846 A114847 this_sequence A114849 A114850 A114851
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
I. Tasoulas (jtas(AT)unipi.gr), Feb 20 2006
|
|
|
Search completed in 0.002 seconds
|
