A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.

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, 18, 1

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

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Peter Bala, Factorisations of some Riordan arrays as infinite products

Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.

Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.

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>=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). - Philippe Deléham, Jan 23 2010

From Gary W. Adamson, Jul 07 2011: (Start)

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, ...). (End)

Working with an offset of 0, the inverse array is the Riordan array ((1 - x)^2, x*(1 - x)^2). - Peter Bala, Apr 30 2024

Example

Triangle begins:
     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;
  ...
Top row of M^3 = (30, 18, 6, 1)
From Peter Bala, Nov 25 2024: (Start)
The transposed array as an infinite product of upper triangular arrays:
  /1 2 3 4 5 ... \/1            \/1              \       /1 2 7 30 143 ...\
  |  1 2 3 4 ... ||  1 2 3 4 ...||  1            |       |  1 4 18  88 ...|
  |    1 2 3 ... ||    1 2 3 ...||    1 2 3 4 ...| ... = |    1  6  33 ...|
  |      1 2 ... ||      1 2 ...||      1 2 3 ...|       |       1   8 ...|
  |        1 ... ||        1 ...||        1 2 ...|       |           1 ...|
  |          ... ||          ...||            ...|       |             ...|
Cf. A078812. (End)

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 *)

Prog

(PARI) T(n, k) = 2*k*binomial(3*n-k, n-k)/(3*n-k); \\ Andrew Howroyd, Nov 06 2017

Crossrefs

Row sums give sequence A001764.

Columns 1..5 are A006013, A006629, A006630, A006631, A233657.

Sequence in context: A072248 A317360 A177011 * A011274 A122843 A167196

Adjacent sequences: A092273 A092274 A092275 * A092277 A092278 A092279

Keyword

nonn,tabl

Author

Emeric Deutsch, Feb 24 2004

Status

approved