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A284417 Triangular array read by rows. T(n,k) is the number of rooted labeled trees on n nodes with exactly k vertices whose unique descendent is a leaf, n >= 1, 0 <= k <= floor((n-1)/2) + delta_{2,n}. 0
1, 0, 2, 3, 6, 16, 48, 145, 420, 60, 1536, 4800, 1440, 19579, 65730, 31500, 840, 290816, 1053696, 698880, 53760, 4942305, 19332936, 16367400, 2388960, 15120, 94689280, 399052800, 410296320, 93542400, 2419200, 2020278931, 9146127870, 11044008360, 3526261200, 200415600, 332640, 47523053568, 230339788800, 319018106880, 133013422080, 12986265600, 127733760 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Column k=0 is A052318(n) for n>2.

Row sums = n^(n-1) = A000169(n).

LINKS

Table of n, a(n) for n=1..43.

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 183.

FORMULA

E.g.f. satisfies: A(x,y) = x exp(A(x,y)) - x^2 + y x^2.

EXAMPLE

Triangle begins

       1,

       0,       2,

       3,       6,

      16,      48,

     145,     420,     60,

    1536,    4800,   1440,

   19579,   65730,  31500,   840,

  290816, 1053696, 698880, 53760,

  ...

T(3,1)=6 because there are 6 labeled rooted trees (paths) o-o-o and these 6 trees have 1 vertex whose only descendent is a leaf. T(3,0) = 3 because there are 3 labeled trees of the form

    o

   / \

  o   o

and these 3 trees have no such vertices.

MATHEMATICA

nn = 10; Range[0, nn]! CoefficientList[Series[-z^2 + u z^2 - ProductLog[-E^((-1 + u) z^2) z], {z, 0, nn}], {z, u}] // Grid

CROSSREFS

Cf. A055302.

Sequence in context: A159342 A089872 A006402 * A219024 A145860 A305190

Adjacent sequences:  A284414 A284415 A284416 * A284418 A284419 A284420

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Mar 26 2017

STATUS

approved

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Last modified August 12 23:19 EDT 2020. Contains 336440 sequences. (Running on oeis4.)