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

%I A092276 #47 Feb 18 2025 08:58:20
%S A092276 1,2,1,7,4,1,30,18,6,1,143,88,33,8,1,728,455,182,52,10,1,3876,2448,
%T A092276 1020,320,75,12,1,21318,13566,5814,1938,510,102,14,1,120175,76912,
%U A092276 33649,11704,3325,760,133,16,1,690690,444015,197340,70840,21252,5313,1078,168,18,1
%N A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.
%C A092276 With offset 0, Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A006013. - _Philippe Deléham_, Jan 23 2010
%H A092276 Andrew Howroyd, <a href="/A092276/b092276.txt">Table of n, a(n) for n = 1..1275</a>
%H A092276 Peter Bala, <a href="/A033184/a033184.pdf">Factorisations of some Riordan arrays as infinite products</a>
%H A092276 Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.
%H A092276 Paul Barry, <a href="https://arxiv.org/abs/2011.13985">The second production matrix of a Riordan array</a>, arXiv:2011.13985 [math.CO], 2020.
%H A092276 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of non-crossing configurations</a>, Discrete Math., 204, 203-229, 1999.
%H A092276 M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00121-0">Enumeration of noncrossing trees on a circle</a>, Discrete Math., 180, 301-313, 1998.
%F A092276 T(n, k) = 2*k*binomial(3n-k, n-k)/(3n-k).
%F A092276 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.
%F A092276 T(n, k) = Sum_{j>=1} j*T(n-1, k-2+j). - _Philippe Deléham_, Sep 14 2005
%F A092276 With offset 0, T(n,k) = ((n+1)/(k+1))*binomial(3n-k+1, n-k). - _Philippe Deléham_, Jan 23 2010
%F A092276 From _Gary W. Adamson_, Jul 07 2011: (Start)
%F A092276 Let M = the production matrix
%F A092276   2, 1;
%F A092276   3, 2, 1;
%F A092276   4, 3, 2, 1;
%F A092276   5, 4, 3, 2, 1;
%F A092276   ...
%F A092276 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)
%F A092276 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
%e A092276 Triangle begins:
%e A092276      1;
%e A092276      2,    1;
%e A092276      7,    4,    1;
%e A092276     30,   18,    6,   1;
%e A092276    143,   88,   33,   8,  1;
%e A092276    728,  455,  182,  52, 10,  1;
%e A092276   3876, 2448, 1020, 320, 75, 12, 1;
%e A092276   ...
%e A092276 Top row of M^3 = (30, 18, 6, 1)
%e A092276 From _Peter Bala_, Nov 25 2024: (Start)
%e A092276 The transposed array as an infinite product of upper triangular arrays:
%e A092276   /1 2 3 4 5 ... \/1            \/1              \       /1 2 7 30 143 ...\
%e A092276   |  1 2 3 4 ... ||  1 2 3 4 ...||  1            |       |  1 4 18  88 ...|
%e A092276   |    1 2 3 ... ||    1 2 3 ...||    1 2 3 4 ...| ... = |    1  6  33 ...|
%e A092276   |      1 2 ... ||      1 2 ...||      1 2 3 ...|       |       1   8 ...|
%e A092276   |        1 ... ||        1 ...||        1 2 ...|       |           1 ...|
%e A092276   |          ... ||          ...||            ...|       |             ...|
%e A092276 Cf. A078812. (End)
%p A092276 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);
%t A092276 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 *)
%o A092276 (PARI) T(n, k) = 2*k*binomial(3*n-k, n-k)/(3*n-k); \\ _Andrew Howroyd_, Nov 06 2017
%Y A092276 Row sums give sequence A001764.
%Y A092276 Columns 1..5 are A006013, A006629, A006630, A006631, A233657.
%K A092276 nonn,tabl
%O A092276 1,2
%A A092276 _Emeric Deutsch_, Feb 24 2004