A236396 - OEIS

%I #17 Aug 17 2017 15:43:31

%S 1,0,2,0,3,9,0,4,40,64,0,5,205,505,625,0,6,1176,4536,7056,7776,0,7,

%T 7399,46249,89929,112609,117649,0,8,50576,526352,1284032,1835072,

%U 2056832,2097152,0,9,372537,6604497,20351601,33188481,40325121,42683841,43046721

%N 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 If we replace each row by its differences we get A034855.

%H Alois P. Heinz, <a href="/A236396/b236396.txt">Rows n = 1..100, flattened</a>

%H 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 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 Triangle begins:

%e [1],

%e [0, 2],

%e [0, 3, 9],

%e [0, 4, 40, 64],

%e [0, 5, 205, 505, 625],

%e [0, 6, 1176, 4536, 7056, 7776],

%e [0, 7, 7399, 46249, 89929, 112609, 117649],

%e [0, 8, 50576, 526352, 1284032, 1835072, 2056832, 2097152],

%e ...

%p gf:= proc(k) gf(k):= `if`(k=0, x, x*exp(gf(k-1))) end:

%p A:= proc(n, k) A(n, k):= n!*coeff(series(gf(k), x, n+1), x, n) end:

%p [seq([seq(A(n, d), d=0..n-1)], n=1..12)];

%t 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 Cf. A034855, A001854, A235595, A234953.

%K nonn,tabl

%O 1,3

%A _N. J. A. Sloane_, Jan 28 2014