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A236396 Triangle read by rows: T(n,k) = number of rooted labeled trees with n nodes and height <= k, for n >= 1, 0 <= k <= n-1.

%I A236396 #17 Aug 17 2017 15:43:31
%S A236396 1,0,2,0,3,9,0,4,40,64,0,5,205,505,625,0,6,1176,4536,7056,7776,0,7,
%T A236396 7399,46249,89929,112609,117649,0,8,50576,526352,1284032,1835072,
%U A236396 2056832,2097152,0,9,372537,6604497,20351601,33188481,40325121,42683841,43046721
%N A236396 Triangle read by rows: T(n,k) = number of rooted labeled trees with n nodes and height <= k, for n >= 1, 0 <= k <= n-1.
%C A236396 If we replace each row by its differences we get A034855.
%H A236396 Alois P. Heinz, <a href="/A236396/b236396.txt">Rows n = 1..100, flattened</a>
%H A236396 J. Riordan, <a href="http://www.research.ibm.com/journal/rd/045/ibmrd0405E.pdf">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev. 4 (1960), 473-478. [broken link]
%H A236396 J. Riordan, <a href="http://dx.doi.org/10.1147/rd.45.0473">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev. 4 (1960), 473-478.
%e A236396 Triangle begins:
%e A236396 [1],
%e A236396 [0, 2],
%e A236396 [0, 3, 9],
%e A236396 [0, 4, 40, 64],
%e A236396 [0, 5, 205, 505, 625],
%e A236396 [0, 6, 1176, 4536, 7056, 7776],
%e A236396 [0, 7, 7399, 46249, 89929, 112609, 117649],
%e A236396 [0, 8, 50576, 526352, 1284032, 1835072, 2056832, 2097152],
%e A236396 ...
%p A236396 gf:= proc(k) gf(k):= `if`(k=0, x, x*exp(gf(k-1))) end:
%p A236396 A:= proc(n, k) A(n, k):= n!*coeff(series(gf(k), x, n+1), x, n) end:
%p A236396 [seq([seq(A(n, d), d=0..n-1)], n=1..12)];
%t A236396 gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k-1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; Table[Table[a[n, d], {d, 0, n-1}], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Mar 07 2014, after Maple *)
%Y A236396 Cf. A034855, A001854, A235595, A234953.
%K A236396 nonn,tabl
%O A236396 1,3
%A A236396 _N. J. A. Sloane_, Jan 28 2014