1,3

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).

M. Bousqet-Melou, 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.

T(n, k)=sum([(k+i)/(2*n-k+i)] binomial(k-1, i) binomial(3n-2k+i-1, n-k), i=0..k-1).

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).

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;

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

Cf. A001764.

Sequence in context: A084933 A138153 A023395 this_sequence A131401 A061446 A107476

Adjacent sequences: A101406 A101407 A101408 this_sequence A101410 A101411 A101412

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 15 2005 and Jan 17 2005

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 25 2009 at the suggestion of R. J. Mathar and Max Alekseyev.