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A101409 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the leftmost leaf is at level k. 0
1, 1, 2, 3, 5, 4, 12, 19, 16, 8, 55, 85, 73, 44, 16, 273, 416, 361, 234, 112, 32, 1428, 2156, 1883, 1269, 680, 272, 64, 7752, 11628, 10200, 7043, 4016, 1856, 640, 128, 43263, 64581, 56829, 39897, 23665, 11864, 4848, 1472, 256, 246675, 366850, 323587, 229936, 140161, 74050, 33360, 12256, 3328, 512 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,k) is also the number of diagonally convex directed polyominoes with n diagonals and having k diagonals of length 1. Proof: the two triangles have the same g.f.

Row n has n terms. Column 1 and row sums yield the ternary numbers (A001764).

LINKS

Table of n, a(n) for n=1..55.

M. Bousquet-Mélou, Percolation models and animals, Europ. J. Combinatorics, 17, 1996, 343-369 (Prop. 2.4).

E. Deutsch, S. Feretic and M. Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., 256 (2002), 645-654.

FORMULA

T(n,k) = Sum_{i=0..k-1}((k+i)/(2*n-k+i)) binomial(k-1, i) binomial(3n-2k+i-1, n-k).

G.f. = (1-tzg^2)/(1-tzg-tzg^2), where g=1+zg^3 is the g.f. of the ternary numbers (A001764). (An explicit expression for g is given in the Maple program.)

EXAMPLE

T(2,1)=1 and T(2,2)=2 because the noncrossing trees with 2 edges are /\, /_ and _\.

Or, T(2,2)=2 because the horizontal domino and the vertical domino have 2 diagonals of length 1 each.

Triangle begins:

   1;

   1,  2;

   3,  5,  4;

  12, 19, 16,  8;

  55, 85, 73, 44, 16;

MAPLE

G:=t*z*g/(1-t*z*g-t*z*g^2): g:=2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z): Gser:=simplify(series(G, z=0, 12)): Gser:=simplify(series(G, z=0, 14)): for n from 1 to 10 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 1 to 10 do seq(coeff(P[n], t^k), k=1..n) od;

T:=proc(n, k) if k=1 then binomial(3*n-3, n-1)/(2*n-1) elif k<=n then sum(((k+i)/(2*n-k+i))*binomial(k-1, i)*binomial(3*n-2*k+i-1, n-k), i=0..k-1) else 0 fi end: for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

MATHEMATICA

T[n_, k_] := Sum[(k+i)/(2n-k+i) Binomial[k-1, i] Binomial[3n-2k+i-1, n-k], {i, 0, k-1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 18 2017 *)

CROSSREFS

Cf. A001764.

Sequence in context: A171038 A023395 A193798 * A271862 A131401 A061446

Adjacent sequences:  A101406 A101407 A101408 * A101410 A101411 A101412

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jan 15 2005 and Jan 17 2005

EXTENSIONS

Edited by N. J. A. Sloane, Jan 25 2009 at the suggestion of R. J. Mathar and Max Alekseyev

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.