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A007151 Number of planted evolutionary trees of magnitude n.
(Formerly M3064)
4
1, 3, 19, 198, 2906, 55018, 1275030, 34947664, 1105740320, 39661089864, 1590232358584, 70482038536880, 3421732373367504, 180574681050278960, 10292371442183694832, 630125771602386523392, 41239934114630205030656 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also number of labeled rooted trees with n generators. (A generator is a leaf or a node with just one child.) - Christian G. Bower, Jun 07 2005

REFERENCES

L. R. Foulds and R. W. Robinson, Counting certain classes of evolutionary trees with singleton labels, Congress. Num., 44 (1984), 65-88.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..360

L. R. Foulds and R. W. Robinson, Counting certain classes of evolutionary trees with singleton labels, Congress. Num., 44 (1984), 65-88. (Annotated scanned copy)

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

FORMULA

E.g.f. satisfies (2-x)*A(x) = x - 1 + exp(A(x)). - Christian G. Bower, Jun 07 2005

a(n) = Sum_{k=1..(n-1)} (n+k-1)!*Sum_{j=1..k} 1/((k-j)!)*Sum_{i=0..(n-1)} binomial(j+i-1,j-1)*Sum_{m=0..j} 2^m*(-1)^(m+i)*Stirling2(n-m+j-i-1,j-m))/(m!*(n-m+j-i-1)!), n>1, a(1)=1. - Vladimir Kruchinin, Aug 07 2012

a(n) ~ sqrt(LambertW(1)+1) * n^(n-1) * (LambertW(1))^n / (exp(n) * (2*LambertW(1)-1)^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014

MAPLE

A007151 := proc(n)

    local k, j, i, m , a;

    if n =1 then

        1;

    else

        a := 0 ;

        for k from 1 to n-1 do

        for j from 1 to k do

        for i from 0 to n-1 do

        for m from 0 to j do

             a := a+(n+k-1)! /(k-j)! *binomial(j+i-1, j-1) *2^m *(-1)^(m+i) *combinat[stirling2](n-m+j-i-1, j-m) / m! /(n-m+j-i-1)! ;

        end do:

        end do:

        end do:

        end do:

        a ;

    end if;

end proc:

seq(A007151(n), n=1..10) ; # R. J. Mathar, Mar 19 2018

MATHEMATICA

Rest[CoefficientList[InverseSeries[Series[(1 - E^x + 2*x)/(1 + x), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

PROG

(Maxima) a(n):=if n=1 then 1 else (sum((n+k-1)!*sum(1/((k-j)!)*sum(binomial(j+i-1, j-1)*sum((2^m*(-1)^(m+i)*stirling2(n-m+j-i-1, j-m))/(m!*(n-m+j-i-1)!), m, 0, j), i, 0, n-1), j, 1, k), k, 1, n-1)); /* Vladimir Kruchinin, Aug 07 2012 */

(PARI) for(n=1, 20, print1(if(n==1, 1, sum(k=1, n-1, (n+k-1)!*sum(j=1, k, (1/(k-j)!)* sum(i=0, n-1, binomial(j+i-1, j-1)*sum(m=0, j, 2^m*(-1)^(m+i)* stirling(n-m+j-i-1, j-m, 2)/(m!*(n-m+j-i-1)!)))))), ", ")) \\ G. C. Greubel, Nov 26 2017

CROSSREFS

Cf. A007152, A108521, A108522, A000169, A000311.

Sequence in context: A001832 A195511 A123681 * A269787 A303927 A288693

Adjacent sequences:  A007148 A007149 A007150 * A007152 A007153 A007154

KEYWORD

nonn,easy

AUTHOR

Robert W. Robinson

STATUS

approved

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Last modified November 15 13:47 EST 2019. Contains 329149 sequences. (Running on oeis4.)