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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}.

%I A284417 #25 Jan 10 2018 03:35:49
%S A284417 1,0,2,3,6,16,48,145,420,60,1536,4800,1440,19579,65730,31500,840,
%T A284417 290816,1053696,698880,53760,4942305,19332936,16367400,2388960,15120,
%U A284417 94689280,399052800,410296320,93542400,2419200,2020278931,9146127870,11044008360,3526261200,200415600,332640,47523053568,230339788800,319018106880,133013422080,12986265600,127733760
%N 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}.
%C A284417 Column k=0 is A052318(n) for n>2.
%C A284417 Row sums = n^(n-1) = A000169(n).
%H A284417 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 183.
%F A284417 E.g.f. satisfies: A(x,y) = x exp(A(x,y)) - x^2 + y x^2.
%e A284417 Triangle begins
%e A284417        1,
%e A284417        0,       2,
%e A284417        3,       6,
%e A284417       16,      48,
%e A284417      145,     420,     60,
%e A284417     1536,    4800,   1440,
%e A284417    19579,   65730,  31500,   840,
%e A284417   290816, 1053696, 698880, 53760,
%e A284417   ...
%e A284417 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
%e A284417     o
%e A284417    / \
%e A284417   o   o
%e A284417 and these 3 trees have no such vertices.
%t A284417 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
%Y A284417 Cf. A055302.
%K A284417 nonn,tabf
%O A284417 1,3
%A A284417 _Geoffrey Critzer_, Mar 26 2017