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Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having k UDUD's (here U=(1,1), D=(1,-1)).
6

%I #30 May 12 2018 22:17:12

%S 1,1,1,1,3,1,1,7,5,1,1,19,14,7,1,1,53,46,22,9,1,1,153,150,82,31,11,1,

%T 1,453,495,299,127,41,13,1,1,1367,1651,1087,507,181,52,15,1,1,4191,

%U 5539,3967,1991,781,244,64,17,1,1,13015,18692,14442,7824,3271,1128,316,77,19

%N Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having k UDUD's (here U=(1,1), D=(1,-1)).

%C Column k=0 is A078481.

%C Column k=1 is A244236.

%C Row sums are the Catalan numbers (A000108).

%H Alois P. Heinz, <a href="/A094507/b094507.txt">Rows n = 0..150, flattened</a>

%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 G.f.: G=G(t, z) satisfies the equation z(1+z-tz)G^2-(1+z+z^2-tz-tz^2)G+1+z-tz=0.

%e T(3,0) = 3 because UDUUDD, UUDDUD and UUUDDD are the only Dyck paths of semilength 3 and having no UDUD in them.

%e Triangle begins:

%e 1;

%e 1;

%e 1, 1;

%e 3, 1, 1;

%e 7, 5, 1, 1;

%e 19, 14, 7, 1, 1;

%e 53, 46, 22, 9, 1, 1;

%e 153, 150, 82, 31, 11, 1, 1;

%e 453, 495, 299, 127, 41, 13, 1, 1;

%e 1367, 1651, 1087, 507, 181, 52, 15, 1, 1;

%e 4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1;

%p b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,

%p `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 2][t])

%p +b(x-1, y-1, [1, 3, 1, 3][t])*`if`(t=4, z, 1))))

%p end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):

%p seq(T(n), n=0..15); # _Alois P. Heinz_, Jun 02 2014

%t b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, {2, 2, 4, 2}[[t]]] + b[x-1, y-1, {1, 3, 1, 3}[[t]]]*If[t == 4, z, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1] ]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Apr 29 2015, after _Alois P. Heinz_ *)

%Y Cf. A078481, A000108, A102405 (the same for DUDU), A243752, A243753, A244236.

%Y T(2n,n) gives A304361.

%K nonn,tabf

%O 0,5

%A _Emeric Deutsch_, Jun 05 2004