1,2

Also called sagittal graphs.

T(n,k)=1 iff n=k (counts the identity mapping of [n]) - Len Smiley (smiley(AT)math.uaa.alaska.edu), Apr 03 2006

Also the coefficients of the tree polynomials t_{n}(y) defined by (1-T(z))^(-y) = Sum_{n>=0} t_{n}(y) (z^n/n!) where T(z) is Cayley's tree function T(z) = Sum_{n>=1} n^(n-1) (z^n/n!) giving the number of labeled trees A000169. [From Peter Luschny (peter(AT)luschny.de), Mar 03 2009]

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Julia Handl and Joshua Knowles, An Investigation of Representations and Operators for Evolutionary Data Clustering with a Variable Number of Clusters, in Parallel Problem Solving from Nature-PPSN IX, Lecture Notes in Computer Science, Volume 4193/2006, Springer-Verlag. [From N. J. A. Sloane, Jul 09 2009]

D. E. Knuth and B. Pittel. A recurrence related to trees. Proceedings of the American Mathematical Society, 105(2):335-349, 1989. [From Peter Luschny (peter(AT)luschny.de), Mar 03 2009]

J. Riordan, Enumeration of Linear Graphs for Mappings of Finite Sets, Ann. Math. Stat., 33, No. 1, Mar. 1962, pp. 178-185

W. Szpankowski. Average case analysis of algorithms on sequences. John Wiley & Sons, 2001. [From Peter Luschny (peter(AT)luschny.de), Mar 03 2009]

E.g.f.: 1/(1+LambertW(-x))^y.

T(n,k)=sum(j=0..n-1) [binomial(n-1,j)*n^(n-1-j)*(-1)^(k+j+1)*A008275(j+1,k)]=sum(j=0..n-1) [binomial(n-1,j)*n^(n-1-j)*s(j+1,k)]--(Note:s(m,p) denotes signless Stirling cycle number (first kind), A008275 isthe signed triangle) - Len Smiley (smiley(AT)math.uaa.alaska.edu), Apr 03 2006

[1], [3, 1], [17, 9, 1], [142, 95, 18, 1], [1569, 1220, 305, 30, 1], ...

T(3,2)=9: (1,2,3)--> [(2,1,3),(3,2,1),(1,3,2),(1,1,3),(1,2,1), (1,2,2),(2,2,3),(3,2,3),(1,3,3)]

Contribution from Peter Luschny (peter(AT)luschny.de), Mar 03 2009: (Start)

Tree polynomials (with offset 0):

t_0(y) = 1;

t_1(y) = y;

t_2(y) = 3y + y^2;

t_3(y) = 17y + 9y^2 + y^3; (End)

with(combinat):T:=array(1..8, 1..8):for m from 1 to 8 do for p from 1 to m do T[m, p]:=sum(binomial(m-1, k)*m^(m-1-k)*(-1)^(p+k+1)*stirling1(k+1, p), k=0..m-1); print(T[m, p]) od od; - Len Smiley (smiley(AT)math.uaa.alaska.edu), Apr 03 2006

Contribution from Peter Luschny (peter(AT)luschny.de), Mar 03 2009: (Start)

T := z -> sum(n^(n-1)*z^n/n!, n=1..16):

p := convert(simplify(series((1-T(z))^(-y), z, 12)), 'polynom'):

seq(print(coeff(p, z, i)*i!), i=0..8); (End)

Row sums: A000312.

Cf. k=1: A001865 ; k=2: A065456.

Sequence in context: A123527 A096611 A162313 this_sequence A151918 A089974 A143849

Adjacent sequences: A060278 A060279 A060280 this_sequence A060282 A060283 A060284

easy,nonn,tabl

Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 09 2001