

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.


1



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
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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

Andrew Howroyd, Table of n, a(n) for n = 1..1275
M. BousquetMélou, Percolation models and animals, Europ. J. Combinatorics, 17, 1996, 343369 (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), 645654.


FORMULA

T(n,k) = Sum_{i=0..k1}((k+i)/(2*nk+i)) binomial(k1, i) binomial(3n2k+i1, nk).
G.f. = (1tzg^2)/(1tzgtzg^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/(1t*z*gt*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*n3, n1)/(2*n1) elif k<=n then sum(((k+i)/(2*nk+i))*binomial(k1, i)*binomial(3*n2*k+i1, nk), i=0..k1) 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)/(2nk+i) Binomial[k1, i] Binomial[3n2k+i1, nk], {i, 0, k1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* JeanFrançois Alcover, Mar 18 2017 *)


PROG

(PARI) T(n, k)={sum(i=0, k1, ((k+i)/(2*nk+i))*binomial(k1, i)*binomial(3*n2*k+i1, nk))} \\ Andrew Howroyd, Nov 17 2017


CROSSREFS

Cf. A001764.
Sequence in context: A316655 A318848 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



