A135330 - OEIS

%I #14 Nov 16 2019 16:22:19

%S 1,1,2,4,1,10,4,28,14,85,46,1,271,151,7,893,502,35,3013,1697,151,1,

%T 10351,5828,607,10,36075,20293,2353,65,127219,71494,8952,346,1,453097,

%U 254404,33738,1648,13,1627378,913028,126594,7336,104

%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k UUDU's starting at level 0.

%C Row n has 1+floor(n/3) terms. Row sums yield the Catalan numbers (A000108). Column 0 yields A135336. - _Emeric Deutsch_, Dec 14 2007

%H A. Sapounakis, I. Tasoulas and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.005">Counting strings in Dyck paths</a>, Discrete Math., 307 (2007), 2909-2924.

%F From _Emeric Deutsch_, Dec 14 2007: (Start)

%F T(n,k) = (1/(n+1))*Sum_{j=k..floor(n/3)} (-1)^(j-k)*(3j+1)*binomial(j,k)*binomial(2n-3j, n).

%F G.f.: C/(1 + (1-t)z^3*C^3), where C = (1-sqrt(1-4z))/(2z) is the g.f. of the Catalan numbers (A000108). (End)

%e Triangle begins:

%e       1;

%e       1;

%e       2;

%e       4,    1;

%e      10,    4;

%e      28,   14;

%e      85,   46,   1;

%e     271,  151,   7;

%e     893,  502,  35;

%e    3013, 1697, 151,  1;

%e   10351, 5828, 607, 10;

%e   ...

%e T(4,1)=4 because we have UUDUDDUD, UUDUUDDD, UUDUDUDD and UDUUDUDD.

%p T:=proc(n,k) options operator, arrow: (sum((-1)^(j-k)*(3*j+1)*binomial(j,k)*binomial(2*n-3*j,n),j=k..floor((1/3)*n)))/(n+1) end proc: for n from 0 to 14 do seq(T(n,k),k=0..floor((1/3)*n)) end do; # yields sequence in triangular form; _Emeric Deutsch_, Dec 14 2007

%Y Cf. A000108, A135336.

%K nonn,tabf

%O 0,3

%A _N. J. A. Sloane_, Dec 07 2007

%E Edited and extended by _Emeric Deutsch_, Dec 14 2007