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 A091187 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k branches. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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. 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. A001006, A000108. Cf. A007476. [Gary W. Adamson, Dec 31 2008] Sequence in context: A293472 A046726 A082137 * A318607 A340106 A259824 Adjacent sequences:  A091184 A091185 A091186 * A091188 A091189 A091190 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Feb 23 2004 STATUS approved

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Last modified April 10 21:34 EDT 2021. Contains 342856 sequences. (Running on oeis4.)