|
Search: id:A114596
|
|
|
| A114596 |
|
Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having abscissa of first return equal to 2k (2<=k<=n). A hill in a Dyck path is a peak at level 1. |
|
+0
1
|
|
| 1, 0, 2, 1, 0, 5, 2, 2, 0, 14, 6, 4, 5, 0, 42, 18, 12, 10, 14, 0, 132, 57, 36, 30, 28, 42, 0, 429, 186, 114, 90, 84, 84, 132, 0, 1430, 622, 372, 285, 252, 252, 264, 429, 0, 4862, 2120, 1244, 930, 798, 756, 792, 858, 1430, 0, 16796, 7338, 4240, 3110, 2604, 2394, 2376
(list; graph; listen)
|
|
|
OFFSET
|
2,3
|
|
|
COMMENT
|
Row sums are the Fine numbers (A000957). Column 2 yield the Fine numbers (A000957). T(n,n)=Catalan(n-1) (A000108). Sum(k*T(n,k),k=2..n)=2*A014301(n).
|
|
REFERENCES
|
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241, 2001, 241-265.
|
|
FORMULA
|
T(n, k)=c(k-1)f(n-k) (2<=k<=n), where c(n)=binomial(2n, n)/(n+1) (the Catalan numbers; A000108) and f(n)=3sum(binomial(2n-2j, n), j=0..floor(n/2))-binomial(2n+2, n+1) (the Fine numbers, A000957). G.f. =[2+2z-2tz+sqrt(1-4z)-sqrt(1-4tz)]/[1+2z+sqrt(1-4z)]-1.
|
|
EXAMPLE
|
T(5,3)=2 because we have UUUDDD|UUDD and UUDUDD|UUDD, where U=(1,1), D=(1,-1) (first return is shown by a vertical bar).
Triangle begins:
1;
0,2;
1,0,5;
2,2,0,14;
6,4,5,0,42;
18,12,10,14,0,132;
|
|
MAPLE
|
c:=n->binomial(2*n, n)/(n+1): f:=n->3*sum(binomial(2*n-2*j, n), j=0..floor(n/2))-binomial(2*n+2, n+1): for n from 2 to 12 do seq(c(k-1)*f(n-k), k=2..n) od; # yields sequence in triangular form
|
|
CROSSREFS
|
Cf. A000957, A000108, A014301.
Sequence in context: A133727 A103185 A130513 this_sequence A083417 A021479 A073583
Adjacent sequences: A114593 A114594 A114595 this_sequence A114597 A114598 A114599
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 12 2005
|
|
|
Search completed in 0.002 seconds
|
