A114608 Triangle read by rows: T(n,k) is the number of bicolored Dyck paths of semilength n and having k peaks of the form ud (0 <= k <= n). A bicolored Dyck path is a Dyck path in which each up-step is of two kinds: u and U.

1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 45, 96, 66, 16, 1, 197, 501, 450, 170, 25, 1, 903, 2668, 2955, 1520, 365, 36, 1, 4279, 14407, 18963, 12355, 4165, 693, 49, 1, 20793, 78592, 119812, 94528, 41230, 9856, 1204, 64, 1, 103049, 432073, 748548, 693588, 372078, 117054

Offset: 0

Author

Emeric Deutsch, Dec 15 2005

Keywords

• nonn
• tabl

Comments

Row sums yield A052701. Column 0 yields the little Schroeder numbers (A001003). Sum_{k=0..n} k*T(n,k) = A069720(n).

Triangle T(n,k), 0 <= k <= n, read by rows; given by [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 23 2005

Examples

			T(3,2)=9 because we have (ud)(ud)Ud, (ud)Ud(ud), Ud(ud)(ud), (ud)u(ud)d,
(ud)U(ud)d, u(ud)d(ud), U(ud)d(ud), u(ud)(ud)d and U(ud)(ud)d (the ud peaks are shown between parentheses).
Triangle starts:
   1;
   1,  1;
   3,  4,  1;
  11, 19,  9,  1;
  45, 96, 66, 16,  1;
		

Links

• Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.

Crossrefs

Cf. A001003, A052701, A069720.

Programs

Maple

T:=proc(n,k) if k<=n-1 then (1/n)*binomial(n,k)*sum(2^j*binomial(n,j+1)*binomial(n-k,j),j=0..n-k) elif k=n then 1 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

T[n_, k_] := If[k <= n-1, (1/n)*Binomial[n, k]*Sum[2^j*Binomial[n, j+1]* Binomial[n-k, j], {j, 0, n-k}], If[k == n, 1, 0]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 11 2018, from Maple *)

Formula

T(n,k) = (1/n)*binomial(n,k)*Sum_{j=0..n-k} 2^j*binomial(n, j+1)*binomial(n-k, j) (k <= n-1); T(n, n)=1.

G.f. = G = G(t, z) satisfies G = 1 + z*(G-1+t)*G + z*G^2.