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A091187
Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k branches.
3
1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 16, 9, 1, 5, 20, 40, 45, 21, 1, 6, 30, 80, 135, 126, 51, 1, 7, 42, 140, 315, 441, 357, 127, 1, 8, 56, 224, 630, 1176, 1428, 1016, 323, 1, 9, 72, 336, 1134, 2646, 4284, 4572, 2907, 835, 1, 10, 90, 480, 1890, 5292, 10710, 15240, 14535, 8350, 2188
OFFSET
1,5
COMMENTS
Row sums are the Catalan numbers A000108. Diagonal entries are the Motzkin numbers A001006.
Equals binomial transform of an infinite lower triangular matrix with A001006 as the main diagonal and the rest zeros. [Gary W. Adamson, Dec 31 2008] [Corrected by Paul Barry, Mar 06 2011]
Reversal of A091869. Diagonal sums are A026418(n+2). [Paul Barry, Mar 06 2011]
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
J.-L. Baril, S. Kirgizov, The pure descent statistic on permutations, Preprint, 2016.
J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combinat. Theory, Ser A, 19, 214-222, 1975.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.
FORMULA
T(n,k) = M(k-1)*binomial(n-1, k-1), where M(k) = A001006(k) = (Sum_{q=0..ceiling((k+1)/2)} binomial(k+1, q)*binomial(k+1-q, q-1))/(k+1) is a Motzkin number.
G.f.: G = G(t,z) satisfies t*z*G^2 -(1 - z + t*z)*G + 1- z + t*z = 0.
From Paul Barry, Mar 06 2011: (Start)
G.f.: 1/(1-x-xy-x^2y^2/(1-x-xy-x^2y^2/(1-x-xy-x^2y^2/(1-... (continued fraction).
G.f.: (1-x(1+y)-sqrt(1-2x(1+y)+x^2(1+2y-3y^2)))/(2x^2*y^2).
E.g.f.: exp(x(1+y))*Bessel_I(1,2*x*y)/(x*y). (End)
EXAMPLE
Triangle begins:
1;
1, 1;
1, 2, 2;
1, 3, 6, 4;
1, 4, 12, 16, 9;
1, 5, 20, 40, 45, 21;
1, 6, 30, 80, 135, 126, 51;
1, 7, 42, 140, 315, 441, 357, 127;
MAPLE
M := n->sum(binomial(n+1, q)*binomial(n+1-q, q-1), q=0..ceil((n+1)/2))/(n+1): T := (n, k)->binomial(n-1, k-1)*M(k-1): seq(seq(T(n, k), k=1..n), n=1..13);
MATHEMATICA
(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; t[n_, k_] := m[k - 1]*Binomial[n - 1, k - 1]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2013 *)
CROSSREFS
Cf. A007476. [Gary W. Adamson, Dec 31 2008]
Sequence in context: A293472 A046726 A082137 * A318607 A340106 A259824
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Feb 23 2004
STATUS
approved