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A000554 Number of labeled trees of diameter 3 with n nodes.
(Formerly M4843 N2070)
2
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

Colin Barker, Table of n, a(n) for n = 4..1000

J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.

Index entries for sequences related to trees

Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

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) = n*(n-1)*(2^(n-3) - 1) = 2*A000217(n-1)*A000225(n-3). - Robert G. Wilson v, Jul 01 2007, corrected by Ilya Gutkovskiy, Sep 17 2016

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

E.g.f.: (x*(exp(x)-1))^2/2. - 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

a(n) = (2^n-8)*(n-1)*n/8. - Colin Barker, Sep 18 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

Sequence in context: A213818 A004302 A277106 * A012289 A012583 A009049

Adjacent sequences:  A000551 A000552 A000553 * A000555 A000556 A000557

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Robert G. Wilson v, Jul 01 2007

STATUS

approved

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Last modified February 20 13:37 EST 2020. Contains 332077 sequences. (Running on oeis4.)