OFFSET
0,2
COMMENTS
Row n contains 1+floor(n/2) terms. Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) (the Catalan numbers). T(n,1)=A003517(n). T(n,2)=A003519(n). Sum(k*T(n,k),k>=0)=A008549(n-1). For both downward and upward crossings, see A118920.
Eigenvector is defined by: A119243(n) = Sum_{k=0..[n\2]} T(n,k)*A119243(k). This triangle is closely related to triangle A119245. - Paul D. Hanna, May 10 2006
Column k is the sum of columns 2k and 2k+1 of A039599. - Philippe Deléham, Nov 11 2008
FORMULA
T(n,k)=(2k+1)binomial(2n+2,n-2k)/(n+1). G.f.=G(t,z)=C^2/(1-tz^2*C^4), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
EXAMPLE
T(3,1)=6 because we have ud\dudu,ud\dduu,udud\du,uudd\du,ud\duud and duud\du (the downward crossings of the x-axis are shown by a back-slash \).
Triangle starts:
1;
2;
5,1;
14,6;
42,27,1;
132,110,10;
MAPLE
T:=(n, k)->(2*k+1)*binomial(2*n+2, n-2*k)/(n+1): for n from 0 to 13 do seq(T(n, k), k=0..floor(n/2)) od; # yields sequence in triangular form
PROG
(PARI) T(n, k)=if(n<2*k || k<0, 0, (2*k+1)*binomial(2*n+2, n-2*k)/(n+1)) - Paul D. Hanna, May 10 2006
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, May 06 2006
STATUS
approved