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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. 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 (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 Andrew Howroyd, Table of n, a(n) for n = 1..1275 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 *) PROG (PARI) T(n, k)={sum(i=0, k-1, ((k+i)/(2*n-k+i))*binomial(k-1, i)*binomial(3*n-2*k+i-1, n-k))} \\ Andrew Howroyd, Nov 17 2017 CROSSREFS Cf. A001764. Sequence in context: A316655 A318848 A193798 * A271862 A309373 A131401 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 May 13 05:02 EDT 2021. Contains 343836 sequences. (Running on oeis4.)