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A135328
Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k UDDU's starting at level 1.
2
1, 1, 2, 4, 1, 10, 4, 29, 12, 1, 90, 36, 6, 290, 114, 24, 1, 960, 376, 86, 8, 3246, 1272, 303, 40, 1, 11164, 4380, 1074, 168, 10, 38934, 15293, 3838, 660, 60, 1, 137358, 54012, 13812, 2528, 290, 12, 489341, 192612, 50013, 9584, 1265, 84, 1
OFFSET
0,3
COMMENTS
Each of rows 0, 1, 2 has one term; row n (n >= 1) has ceiling(n/2) terms. Row sums are the Catalan numbers (A000108). Column 0 yields A135334. - Emeric Deutsch, Dec 14 2007
LINKS
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
FORMULA
From Emeric Deutsch, Dec 14 2007: (Start)
T(n,k) = 2*((k+1)/(n+1))*Sum_{j=k..floor((n-1)/2)} (-1)^(j-k)*binomial(j+1, k+1)*binomial(2n-2j-1, n) (n >= 1).
G.f.: 1 + z*C^2/(1 + (1-t)*z^2*C^2), where C = (1-sqrt(1-4z))/(2z) is the g.f. of the Catalan numbers (A000108). (End)
EXAMPLE
Triangle begins:
1;
1;
2;
4 1;
10 4;
29 12 1;
90 36 6;
290 114 24 1;
960 376 86 8;
3246 1272 303 40 1;
...
T(4,1)=4 because we have UDU(UDDU)D, U(UDDU)DUD, U(UDDU)UDD and UUD(UDDU)D (the UDDU's starting at level 1 are shown between parentheses).
MAPLE
T:=proc(n, k) options operator, arrow: (2*k+2)*(sum((-1)^(j-k)*binomial(j+1, k+1)*binomial(2*n-2*j-1, n), j=k..floor((1/2)*n-1/2)))/(n+1) end proc: 1; for n to 13 do seq(T(n, k), k=0..ceil((n-2)*1/2)) end do; # yields sequence in triangular form; Emeric Deutsch, Dec 14 2007
G:=1+z*C^2/(1+(1-t)*z^2*C^2): C:=((1-sqrt(1-4*z))*1/2)/z: Gser:=simplify(series(G, z=0, 16)): for n from 0 to 13 do P[n]:=sort(coeff(Gser, z, n)) end do: 1; for n to 13 do seq(coeff(P[n], t, j), j=0..floor((n-1)*1/2)) end do; # yields sequence in triangular form; Emeric Deutsch, Dec 14 2007
# third 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, `if`(y=1, 1, 0))*
`if`(t=3, z, 1))+b(x-1, y-1, `if`(t in [1, 2], t+1, 0))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
seq(T(n), n=0..14); # Alois P. Heinz, Nov 16 2019
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, If[y == 1, 1, 0]]*
If[t == 3, z, 1]] + b[x - 1, y - 1, If[1 <= t <= 2, t + 1, 0]]]];
T[n_] := CoefficientList[b[2n, 0, 0], z];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Feb 14 2021, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A114506 A114848 A135330 * A355144 A346419 A048941
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Dec 07 2007
EXTENSIONS
Edited and extended by Emeric Deutsch, Dec 14 2007
STATUS
approved