OFFSET
0,3
COMMENTS
Row sums are Catalan numbers A000108.
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924. - From N. J. A. Sloane, May 05 2012
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.
Triangle begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 4, 1;
: 4 : 10, 4;
: 5 : 28, 13, 1;
: 6 : 82, 44, 6;
: 7 : 248, 153, 27, 1;
: 8 : 770, 536, 116, 8;
: 9 : 2440, 1889, 486, 46, 1;
: 10 : 7858, 6696, 1992, 240, 10;
: 11 : 25644, 23849, 8042, 1180, 70, 1;
: 12 : 84618, 85276, 32124, 5552, 430, 12;
MAPLE
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, expand(b(x-1, y+1, [2, 3, 3, 2, 2][t])
*`if`(t=5, z, 1) +b(x-1, y-1, [1, 1, 4, 5, 1][t]))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15); # Alois P. Heinz, Jun 10 2014
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
KEYWORD
nonn,tabf
AUTHOR
I. Tasoulas (jtas(AT)unipi.gr), Feb 20 2006
STATUS
approved