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A092276
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Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.
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5
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| With offset 0, Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A006013. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 23 2010]
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REFERENCES
| P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discrete Math. 204 (1999), 203-229.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
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FORMULA
| T(n, k)=2k*binomial(3n-k, n-k)/(3n-k). G.f. = 1/(1-tzg^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 DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2005
With offset 0, T(n,k)= ((n+1)/(k+1))*binomial(3n-k+1, n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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
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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)
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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);
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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
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 24 2004
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