OFFSET
0,4
LINKS
D. Baccherini, D. Merlini and R. Sprugnoli, Level generating trees and proper Riordan arrays, Applicable Analysis and Discrete Mathematics, 2, 2008, 69-91 (see p. 89). [From Emeric Deutsch, Sep 21 2008]
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
FORMULA
T(n,k) = (2(k+1)/(n-k))*Sum_{j=0..(n-k)/2} binomial((n-k)/2, j)*binomial((n+k)/2+j, (n-k-2)/2) if k < n and n-k is even; T(n,n)=1; T(n,k)=0 if n-k is odd.
G.f.: R(z^2)/(1-tzR(z^2)), where R = 1 + zR + zR^2 = (1-z-sqrt(1-6z+z^2))/(2z) is the g.f. of the large Schroeder numbers.
T(n,k) = T(n-1,k-1) + T(n-1,k+1) + T(n-2,k), T(0,0)=1. - Philippe Deléham, Nov 18 2009
EXAMPLE
T(3,1)=4 because we have HU, UDU, UH and UUD.
Triangle begins:
1;
0, 1;
2, 0, 1;
0, 4, 0, 1;
6, 0, 6, 0, 1;
0, 16, 0, 8, 0, 1;
MAPLE
T:=proc(n, k) if k<n and n-k mod 2 = 0 then (2*(k+1)/(n-k))*sum(binomial((n-k)/2, j)*binomial((n+k)/2+j, (n-k-2)/2), j=0..(n-k)/2) elif k=n then 1 else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
MATHEMATICA
T[0, 0] = 1;
T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + T[n-1, k+1] + T[n-2, k];
T[_, _] = 0;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 31 2024, after Philippe Deléham *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jul 12 2005
STATUS
approved