|
|
A000554
|
|
Number of labeled trees of diameter 3 with n nodes.
(Formerly M4843 N2070)
|
|
3
|
|
|
12, 60, 210, 630, 1736, 4536, 11430, 28050, 67452, 159588, 372554, 859950, 1965840, 4456176, 10026702, 22412970, 49806980, 110100060, 242220594, 530578950, 1157627352, 2516581800, 5452594550, 11777604930, 25367149836, 54492396756, 116769422490, 249644973150
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
4,1
|
|
REFERENCES
|
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = n(n-1)*S2(n-2, 2) where S2(n, k) denotes the Stirling numbers of 2nd kind. - Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001
a(n) = Sum_{k=1..n-3} binomial(n,2)*binomial(n-2,k). The sum gives the number of Prüfer sequences with exactly 2 distinct digits. - Geoffrey Critzer, Sep 17 2016
O.g.f.: 2*x^4*(6 - 24*x + 33*x^2 - 18*x^3 + 4*x^4)/((1 - x)^3*(1 - 2*x)^3). - Ilya Gutkovskiy, Sep 17 2016
|
|
MATHEMATICA
|
f[n_] := n (n - 1)*StirlingS2[n - 2, 2]; Table[ f@n, {n, 4, 29}] (* Robert G. Wilson v, Jul 01 2007 *)
|
|
PROG
|
(PARI) Vec(2*x^4*(6-24*x+33*x^2-18*x^3+4*x^4)/((1-x)^3*(1-2*x)^3) + O(x^40)) \\ Colin Barker, Sep 18 2016
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|