

A048160


Triangle giving T(n,k) = number of (n,k) labeled rooted Greg trees (n >= 1, 0<=k<=n1).


10



1, 2, 1, 9, 10, 3, 64, 113, 70, 15, 625, 1526, 1450, 630, 105, 7776, 24337, 31346, 20650, 6930, 945, 117649, 450066, 733845, 650188, 329175, 90090, 10395, 2097152, 9492289, 18760302, 20925065, 14194180, 5845455, 1351350, 135135, 43046721
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OFFSET

1,2


COMMENTS

An (n,k) rooted Greg tree can be described as a rooted tree with n black nodes and k white nodes where only the black nodes are labeled and the white nodes have at least 2 children.  Christian G. Bower, Nov 15 1999


LINKS

Table of n, a(n) for n=1..37.
C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122128.
C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122128. (Annotated scanned copy)
C. Flight, Letter to N. J. A. Sloane, Nov 1990
D. J. Jeffrey, G. A. Kalugin, N. Murdoch, Lagrange inversion and Lambert W, Preprint 2015.
M. JosuatVergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees


FORMULA

T(n, 0)=n^(n1), T(n, k)=(n+k2)*T(n1, k1)+(2*n+2*k2)*T(n1, k)+(k+1)*T(n1, k+1).
From Peter Bala, Sep 29 2011: (Start)
E.g.f.: compositional inverse with respect to x of t*(exp(x)1) + (1+t)*x*exp(x) = compositional inverse with respect to x of (x  (2+t)*x^2/2! + (3+2*t)*x^3/3!  (4+3*t)*x^4/4! + ...) = x + (2+t)*x^2/2! + (9+10*t+3*t^2)*x^3/3! + ....
The row generating polynomials R(n,t) satisfy the recurrence R(n+1,t) = (1+t)^2*R'(n,t)+n*(2+t)*R(n,t) with R(1,t) = 1.
The shifted row polynomials R(n,t1) are the row generating polynomials of A054589.
(End)
From Peter Bala, Sep 12 2012: (start)
It appears that the entries in column k = 1 are given by T(n,1) = (n+1)^n  2*n^n (checked up to n = 15)  see A176824.
Assuming this, we could then use the recurrence equation to obtain explicit formulas for columns k = 2,3,....
For example, T(n,2) = 1/2*{(n+2)^(n+1)  4*(n+1)^(n+1) + (4*n+3)*n^n}. (End)


EXAMPLE

1;
2, 1;
9, 10, 3;
64, 113, 70, 15; ...


MATHEMATICA

t[n_ /; n >= 1, k_ /; k >= 0] /; 0 <= k <= n1 := t[n, k] = (n+k2) t[n1, k1] + (2n + 2k  2)*t[n1, k] + (k+1) t[n1, k+1]; t[1, 0] = 1; t[_, _] = 0; Flatten[Table[t[n, k], {n, 1, 9}, {k, 0, n1}]] (* JeanFrançois Alcover, Jul 20 2011, after formula *)


CROSSREFS

Row sums give A005264. Cf. A005263, A048159, A052300A052303. A054589.
Sequence in context: A019615 A132744 A059604 * A305178 A295851 A192324
Adjacent sequences: A048157 A048158 A048159 * A048161 A048162 A048163


KEYWORD

nonn,easy,tabl,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000


STATUS

approved



