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%I #102 May 11 2024 14:03:43
%S 1,2,1,9,6,1,64,48,12,1,625,500,150,20,1,7776,6480,2160,360,30,1,
%T 117649,100842,36015,6860,735,42,1,2097152,1835008,688128,143360,
%U 17920,1344,56,1,43046721,38263752,14880348,3306744,459270,40824,2268,72,1
%N Triangle read by rows: T(n, k) is the number of labeled trees on n nodes with maximal node degree k (0 < k < n).
%C Essentially the coefficients of the Abel polynomials (A137452). - _Peter Luschny_, Jun 12 2022
%C This is a formula from Comtet, Theorem F, vol. I, p. 81 (French edition) used in proving Theorem D.
%C If we let N = n+1, binomial(N-2, k-1)*(N-1)^(N-k-1) = binomial(n-1, k-1)*n^(n-k), so this sequence with offset 1,1 also gives the number of rooted forests of k trees over [n]. - _Washington Bomfim_, Jan 09 2008
%C Let S(n,k) be the signed triangle, S(n,k) = (-1)^(n-k)T(n,k), which starts 1, -2, 1, 9, -6, 1, ..., then the inverse of S is the triangle of idempotent numbers A059298. - _Peter Luschny_, Mar 13 2009
%C With offset 1 also number of labeled multigraphs of k components, n nodes, and no cycles except one loop in each component. See link below to have a picture showing the bijection between rooted forests and multigraphs of this kind. (Note that there are no labels in the picture, but the bijection remains true if we label the nodes.) - _Washington Bomfim_, Sep 04 2010
%C With offset 1, T(n,k) is the number of forests of rooted trees on n nodes with exactly k (rooted) trees. - _Geoffrey Critzer_, Feb 10 2012
%C Also the Bell transform of the sequence (n+1)^n (A000169(n+1)) without column 0. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 21 2016
%C Abel polynomials A(n,x) = x*(x+n)^(n-1) satisfy d/dx A(n,x) = n*A(n-1,x+1). - _Michael Somos_, May 10 2024
%D L. Comtet, Analyse Combinatoire, P.U.F., Paris 1970. Volume 1, p 81.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974.
%H G. C. Greubel, <a href="/A061356/b061356.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A. Avron and N. Dershowitz, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.7.699">Cayley's Formula, A Page From The Book</a>, Amer. Math. Monthly, Vol. 123, No. 7, Aug.-Sep. 2016, 699-700 (2).
%H Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x))</a>
%H W. Bomfim, <a href="http://en.wikipedia.org/wiki/File:Bijecao.gif">Bijection between rooted forests and multigraphs without cycles except one loop in each connected component.</a> [From _Washington Bomfim_, Sep 04 2010]
%H J. W. Moon, <a href="http://www.jstor.org/stable/2312668">Another proof of Cayley's formula for counting trees</a>, Amer. Math. Monthly, Vol. 70, No. 8, Oct. 1963, 846-847.
%H Jim Pitman, <a href="http://www.stat.berkeley.edu/tech-reports/457.pdf">Coalescent Random Forests</a>, Technical Report No. 457, Department of Statistics, University of California.
%H Jim Pitman, <a href="https://doi.org/10.1006/jcta.1998.2919">Coalescent Random Forests</a>, Journal of Combinatorial Theory, Series A, Volume 85, Issue 2, February 1999, Pages 165-193.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbelPolynomial.html">Abel Polynomial</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%H J. Zeng, <a href="http://dx.doi.org/10.1023/A:1009809224933">A Ramanujan sequence that refines the Cayley formula for trees</a>, Ramanujan J., 3(1999) 1, 45-54.
%F T(n, k) = binomial(n-2, k-1)*(n-1)^(n-k-1).
%F E.g.f.: (-LambertW(-y)/y)^(x+1)/(1+LambertW(-y)). - _Vladeta Jovovic_
%F From _Peter Bala_, Sep 21 2012: (Start)
%F Let T(x) = Sum_{n >= 0} n^(n-1)*x^n/n! denote the tree function of A000169. E.g.f.: F(x,t) := exp(t*T(x)) - 1 = -1 + {T(x)/x}^t = t*x + t*(2 + t)*x^2/2! + t*(9 + 6*t + t^2)*x^3/3! + ....
%F The compositional inverse with respect to x of (1/t)*F(x,t) is the e.g.f. for a signed version of the row reverse of A028421.
%F The row generating polynomials are the Abel polynomials A(n,x) = x*(x+n)^(n-1) for n >= 1.
%F Define B(n,x) = x^n/(1+n*x)^(n+1) = (-1)^n*A(-n,-1/x) for n >= 1. The k-th column entries are the coefficients in the formal series expansion of x^k in terms of B(n,x). For example, Col. 1: x = B(1,x) + 2*B(2,x) + 9*B(3,x) + 64*B(4,x) + ..., Col. 2: x^2 = B(2,x) + 6*B(3,x) + 48*B(4,x) + 500*B(5,x) + ... Compare with A059297.
%F n-th row sum = A000272(n+1).
%F Row reverse triangle is A139526.
%F The o.g.f.'s for the diagonals of the triangle are the rational functions R(n,x)/(1-x)^(2*n+1), where R(n,x) are the row polynomials of A155163. See below for examples.
%F (End)
%F T(n,m) = C(n,m)*Sum_{k=1..n-m} m^k*T(n-m,k), T(n,n) = 1. - _Vladimir Kruchinin_, Mar 31 2015
%e : 1;
%e : 2, 1;
%e : 9, 6, 1;
%e : 64, 48, 12, 1;
%e : 625, 500, 150, 20, 1;
%e : 7776, 6480, 2160, 360, 30, 1;
%e ...
%e From _Peter Bala_, Sep 21 2012: (Start)
%e O.g.f.'s for the diagonals begin:
%e 1/(1-x) = 1 + x + x^2 + x^3 + ...
%e 2*x/(1-x)^3 = 2 + 6*x + 12*x^3 + ... A002378(n+1)
%e (9+3*x)/(1-x)^5 = 9 + 48*x + 150*x^2 + ... 3*A004320(n+1)
%e The numerator polynomials are the row polynomials of A155163.
%e (End)
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0,...) as column 0 to the triangle.
%p BellMatrix(n -> (n+1)^n, 12); # _Peter Luschny_, Jan 21 2016
%t nn = 7; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; f[list_] := Select[list, # > 0 &]; Map[f, Drop[Range[0, nn]! CoefficientList[Series[Exp[y t], {x, 0, nn}], {x, y}], 1]] // Flatten (* _Geoffrey Critzer_, Feb 10 2012 *)
%t T[n_, m_] := T[n, m] = Binomial[n, m]*Sum[m^k*T[n-m, k], {k, 1, n-m}]; T[n_, n_] = 1; Table[T[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 31 2015, after _Vladimir Kruchinin_ *)
%t Table[Binomial[n - 2, k - 1]*(n - 1)^(n - k - 1), {n, 2, 12}, {k, 1, n - 1}] // Flatten (* _G. C. Greubel_, Nov 12 2017 *)
%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
%t rows = 10;
%t M = BellMatrix[(# + 1)^#&, rows];
%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%o (Maxima) create_list(binomial(n,k)*(n+1)^(n-k),n,0,20,k,0,n); /* _Emanuele Munarini_, Apr 01 2014 */
%o (Sage) # uses[bell_matrix from A264428]
%o # Adds (1,0,0,0,...) as column 0 to the triangle.
%o bell_matrix(lambda n: (n+1)^n, 12) # _Peter Luschny_, Jan 21 2016
%o (PARI) for(n=2,11, for(k=1,n-1, print1(binomial(n-2, k-1)*(n-1)^(n-k-1), ", "))) \\ _G. C. Greubel_, Nov 12 2017
%Y Variant of A137452.
%Y Columns are A000169, A053506, A053507, A053508, A053509.
%Y First diagonal is A002378.
%Y Row sums give A000272.
%Y Cf. A028421, A059297, A139526 (row reverse), A155163, A202017.
%K easy,nonn,tabl
%O 2,2
%A _Olivier Gérard_, Jun 07 2001
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