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