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A091187
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Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k branches.
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3
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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
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OFFSET
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1,5
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COMMENTS
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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]
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LINKS
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FORMULA
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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.
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)
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EXAMPLE
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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;
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MAPLE
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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);
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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