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A054589
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Table related to labeled rooted trees, cycles and binary trees.
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6
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1, 1, 1, 2, 4, 3, 6, 18, 25, 15, 24, 96, 190, 210, 105, 120, 600, 1526, 2380, 2205, 945, 720, 4320, 13356, 26488, 34650, 27720, 10395, 5040, 35280, 128052, 305620, 507430, 575190, 405405, 135135
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OFFSET
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1,4
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COMMENTS
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The left column is (n-1)!, the right column is (2n-3)!!, the total of each row is n^(n-1).
Constant terms of polynomials related to Ramanujan psi polynomials (see Zeng reference).
Differentiating n times the Lambert function W(x) = Sum_{n>=1} n^(n-1)*x^n/n! with respect to x yields (d/dx)^n W(x) = exp(n*W(x))/(1-W(x))^n*R(n,1/(1-W(x))), where R(n,x) is the n-th row polynomial of this triangle. The first few values are R(1,x) = 1, R(2,x) = 1+x, R(3,x) = 2+4*x+3*x^2. The Ramanujan polynomials R(n,x) are strongly x-log-convex [Chen et al.].
Shor and Dumont-Ramamonjisoa have proved independently that the coefficient of x^k in R(n,x) counts rooted labeled trees on n vertices with k improper edges. Drake, Example 1.7.3, gives another combinatorial interpretation for this triangle as counting a family of labeled trees.
(End)
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LINKS
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FORMULA
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The polynomials p_n = Sum a[n, k]x^k satisfy p_1=1 and p_(n+1) = x*x*dp_n/dx+n*(1+x)*p_n.
E.g.f.: series reversion with respect to x of (1-t+(t-1+x*t)*exp(-x)) = x + (1+t)*x^2/2! + (2+4*t+3*t^2)*x^3/3! + ....
The sequence of shifted row polynomials {p_n(1+t)}n>=1 begins [1,2+t,9+10*t+3*t^2,...]. These are the row polynomials of A048160.
(End)
Let f(x) = exp(x)/(1-t*x). The e.g.f. A(x,t) = x + (1+t)*x^2/2! + (2+4*t+3*t^2)*x^3/3! + ... satisfies the autonomous differential equation dA/dx = f(A). The n-th row polynomial (n>=1) equals D^(n-1)(f(x)) evaluated at x = 0, where D is the operator f(x)*d/dx (apply [Dominici, Theorem 4.1]). - Peter Bala, Nov 09 2011
The polynomials (1+t)^(n-1)*p_n(1/(1+t)) are (up to sign) the row polynomials of A042977. - Peter Bala, Jul 23 2012
Let q_n = Sum_{k>=0} a(n,k)*t^(n-k), with q_0 = 1. (So q_1=t, q_2 = t+t^2, and q_3 = 3*t + 4*t^2 + 2*t^3.) Then Sum_{n>=0} q_n*x^n/n! = t - W((t-1-t^2*x)*exp(t-1)), where W is the Lambert function. - Ira M. Gessel, Jan 06 2012
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EXAMPLE
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Triangle begins:
{1},
{1, 1},
{2, 4, 3},
{6, 18, 25, 15},
...
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MATHEMATICA
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p[1] = 1; p[n_] := p[n] = Expand[x^2*D[p[n-1], x] + (n-1)(1+x)p[n-1]]; Flatten[ Table[ CoefficientList[ p[n], x], {n, 1, 8}]] (* Jean-François Alcover, Jul 22 2011 *)
Clear[a];
a[1, 0] = 1;
a[n_, k_] /; k < 0 || k >= n := 0
a[n_, k_] /; 0 <= k <= n - 1 :=
a[n, k] = (n - 1) a[n - 1, k] + (n + k - 2) a[n - 1, k - 1]
Table[a[n, k], {n, 20}, {k, 0, n - 1} (* David Callan, Oct 14 2012 *)
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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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