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A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k. 5
1, 2, 1, 7, 4, 1, 30, 18, 6, 1, 143, 88, 33, 8, 1, 728, 455, 182, 52, 10, 1, 3876, 2448, 1020, 320, 75, 12, 1, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 690690, 444015, 197340, 70840, 21252, 5313, 1078, 168 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

With offset 0, Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A006013. [Philippe Deléham, Jan 23 2010]

LINKS

Table of n, a(n) for n=1..53.

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

FORMULA

T(n, k) = 2*k*binomial(3n-k, n-k)/(3n-k).

G.f. = 1/(1-t*z*g^2), where g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z) is the g.f. of the sequence A001764.

T(n, k) = Sum_{j, j>=1} j*T(n-1, k-2+j) . - Philippe Deléham, Sep 14 2005

With offset 0, T(n,k)= ((n+1)/(k+1))*binomial(3n-k+1, n-k). [From Philippe Deléham, Jan 23 2010]

Let M = the production matrix

2, 1

3, 2, 1

4, 3, 2, 1

5, 4, 3, 2, 1

...

Top row of M^(n-1) generates n-th row terms of triangle A092276. Leftmost terms of each row = A006013 starting (1, 2, 7, 30, 143,...). - Gary W. Adamson, Jul 07 2011

EXAMPLE

1; 2,1; 7,4,1; 30,18,6,1; 143,88,33,8,1;

Top row of M^3 = (30, 18, 6, 1)

MAPLE

T := proc(n, k) if k=n then 1 else 2*k*binomial(3*n-k, n-k)/(3*n-k) fi end: seq(seq(T(n, k), k=1..n), n=1..11);

MATHEMATICA

t[n_, n_] = 1; t[n_, k_] := 2*k*Binomial[3*n-k, n-k]/(3*n-k); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2012, after Maple *)

CROSSREFS

Row sums give sequence A001764.

First column gives sequence A006013.

Sequence in context: A144696 A072248 A177011 * A011274 A122843 A167196

Adjacent sequences:  A092273 A092274 A092275 * A092277 A092278 A092279

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Feb 24 2004

STATUS

approved

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Last modified April 25 08:30 EDT 2017. Contains 285348 sequences.