A114463 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an odd level (0<=k<=floor(n/2)-1 for n>=2; k=0 for n=0,1).

1, 1, 2, 5, 13, 1, 36, 6, 105, 26, 1, 317, 104, 8, 982, 402, 45, 1, 3105, 1522, 225, 10, 9981, 5693, 1052, 69, 1, 32520, 21144, 4698, 412, 12, 107157, 78188, 20319, 2249, 98, 1, 356481, 288340, 85864, 11522, 679, 14, 1195662, 1061520, 356535, 56360, 4230

Offset: 0

Author

Emeric Deutsch, Nov 29 2005

Keywords

• nonn
• tabf

Comments

Row n (n>=2) has floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(kT(n,k),k=0..floor(n/2)-1)=binomial(2n-4,n) (A002694). Column 0 yields A114465.

Examples

			T(5,1) = 6 because we have UUD(UU)DUDDD, UUD(UU)DDUDD, UUD(UU)DDDUD,
UDUUD(UU)DDD, UUDUD(UU)DDD and UUUDD(UU)DDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an odd level are shown between parentheses; note that the fourth path has an ascent of length 2 that starts at an even level).
Triangle starts:
:  0 :    1;
:  1 :    1;
:  2 :    2;
:  3 :    5;
:  4 :   13,    1;
:  5 :   36,    6;
:  6 :  105,   26,    1;
:  7 :  317,  104,    8;
:  8 :  982,  402,   45,  1;
:  9 : 3105, 1522,  225, 10;
: 10 : 9981, 5693, 1052, 69, 1;
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A114462, A114464, A114465, A000108, A002694, A243752.

Programs

Maple

G:=-1/2*(1-z^2+z^2*t-sqrt((z^2*t-z^2+4*z-1)*(z^2*t-z^2-1)))/z/(-z^2+z^2*t+z-z*t-1): Gser:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 15 do P[n]:=coeff(Gser,z^n) od: 1; 1; for n from 2 to 15 do seq(coeff(t*P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 2, 5, 2][t])
      *`if`(t=5, z, 1) +b(x-1, y-1, [1, 3, 4, 1, 3][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Jun 10 2014

Mathematica

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, 2, 5, 2}[[t]]]*If[t==5, z, 1] + b[x-1, y-1, {1, 3, 4, 1, 3}[[t]]]]]]; 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, Mar 31 2015, after Alois P. Heinz *)

Formula

G.f.: G=G(t, z) satisfies z[(1-t)z^2-(1-t)z+1]G^2-[1-(1-t)z^2]G+1=0.