1,2

Define the matrix function matexps(M) to be exp(M)/exp(1). Then the number of k-level labeled rooted trees with n leaves is also column 0 of the triangle resulting from the n-th iteration of matexps on the Pascal matrix P, A007318. The resulting triangle is also S^n*P*S^-n, where S is the Stirling2 matrix A048993. This function can be coded in PARI as sum(k=0,200,1./k!*M^k)/exp(1)), using exp(M) does not work. See A056857, which equals (1/e)*exp(P) or S*P*S^-1. [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Aug 19 2009]

If we form a table from the family of sequences defined by:

number of k-level labeled rooted trees with n leaves,

then this sequence equals the diagonal in that table:

n=1:A000012=[1,1,1,1,1,1,1,1,1,1,...];

n=2:A000110=[1,2,5,15,52,203,877,4140,21147,115975,...];

n=3:A000258=[1,3,12,60,358,2471,19302,167894,1606137,...];

n=4:A000307=[1,4,22,154,1304,12915,146115,1855570,26097835,...];

n=5:A000357=[1,5,35,315,3455,44590,660665,11035095,204904830,...];

n=6:A000405=[1,6,51,561,7556,120196,2201856,45592666,1051951026,...];

n=7:A001669=[1,7,70,910,14532,274778,5995892,148154860,4085619622,...];

n=8:A081624=[1,8,92,1380,25488,558426,14140722,406005804,13024655442,...];

n=9:A081629=[1,9,117,1989,41709,1038975,29947185,979687005,35839643175,..].

Row n in the above table equals column 0 of matrix power A008277^n

where A008277 = triangle of Stirling numbers of 2nd kind:

1;

1,1;

1,3,1;

1,7,6,1;

1,15,25,10,1;

1,31,90,65,15,1; ...

The name of this sequence is a generalization of the definition given

in the above sequences by Christian G. Bower (bowerc(AT)usa.net).

(PARI) {a(n)=local(E=exp(x+x*O(x^n))-1, F=x); for(i=1, n, F=subst(F, x, E)); n!*polcoeff(F, n)}

Cf. A008277; A000110, A000258, A000307, A000357, A000405, A001669, A081624, A081629.

Sequence in context: A000795 A085628 A053549 this_sequence A130529 A075631 A127182

Adjacent sequences: A139380 A139381 A139382 this_sequence A139384 A139385 A139386

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2008