|
Search: id:A061356
|
|
|
| A061356 |
|
Triangle T(n,k) = labeled trees on n nodes with maximal node degree k (0 < k < n). |
|
+0
2
|
|
| 1, 2, 1, 9, 6, 1, 64, 48, 12, 1, 625, 500, 150, 20, 1, 7776, 6480, 2160, 360, 30, 1, 117649, 100842, 36015, 6860, 735, 42, 1, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1, 43046721, 38263752, 14880348, 3306744, 459270, 40824, 2268, 72, 1
(list; table; graph; listen)
|
|
|
OFFSET
|
2,2
|
|
|
COMMENT
|
This is a formula from Theorem F, vol. I, p. 81 (French edition) used in proving Theorem D.
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 G. Bomfim (webonfim(AT)bol.com.br), Jan 09 2008
|
|
REFERENCES
|
L. Comtet, Analyse Combinatoire, P.U.F., Paris 1970. Volume 1, p 81.
L. Comtet, Advanced Combinatorics, Reidel, 1974.
J. W. Moon, Another proof of Cayley's formula for counting trees, A.M.M., 70 (1963) p846-7.
|
|
LINKS
|
Jim Pitman, Coalescent Random Forests .
J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan J., 3(1999) 1, 45-54.
|
|
FORMULA
|
T(n, k) = binomial(n-2, k-1)*(n-1)^(n-k-1).
E.g.f.: (-LambertW(-y)/y)^(x+1)/(1+LambertW(-y)) (from Vladeta Jovovic (vladeta(AT)Eunet.yu))
|
|
CROSSREFS
|
Columns are A000169, A053506, A053507, A053508, A053509. First diagonal is A002378. Sum of lines gives A000272.
Sequence in context: A103876 A133174 A061691 this_sequence A120671 A086572 A016632
Adjacent sequences: A061353 A061354 A061355 this_sequence A061357 A061358 A061359
|
|
KEYWORD
|
easy,nonn,tabl
|
|
AUTHOR
|
Olivier Gerard (ogerard(AT)ext.jussieu.fr), Jun 07 2001
|
|
|
Search completed in 0.002 seconds
|
