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%I M2732 N1097 #41 Feb 09 2018 21:43:13
%S 0,0,0,1,3,8,18,38,76,147,277,509,924,1648,2912,5088,8823,15170,25935,
%T 44042,74427,125112,209411,348960,579326,958077,1579098,2593903,
%U 4247768,6935070,11290627,18330973,29684082,47946852,77258764,124198083
%N Number of n-node rooted trees of height 3.
%C (1, 1, 2, 3, 5, 8, ...) convolved with (0, 0, 1, 2, 4, 7, ...) = (0, 0, 1, 3, 8, ...). - _Gary W. Adamson_, Aug 14 2010
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A000235/b000235.txt">Table of n, a(n) for n=1..200</a>
%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.
%H J. Riordan, <a href="/A007401/a007401_8.pdf">The enumeration of trees by height and diameter</a>, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
%H N. J. A. Sloane, <a href="/A000235/a000235.txt">Maple programs for counting rooted trees by height (after Riordan)</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F a(n) = A001383(n) - A000041(n-1). - _Christian G. Bower_
%p # For Maple program see link.
%p with(combstruct):
%p ZL:= proc(m) local i; [T0, {seq(T||i=Prod(Z, Set(T||(i+1))), i=0..m-1), T||m=Z}, unlabeled] end: A000235:= n-> count(ZL(3), size=n)-count(ZL(2), size=n): seq(A000235(n), n=1..36); # _Zerinvary Lajos_, Sep 23 2007
%t m = 36; Rest @ CoefficientList[ Series[x*Product[(1-x^k)^(-PartitionsP[k-1]), {k, 1, m}], {x, 0, m}], x] - PartitionsP[Range[0, m-1]] (* _Jean-François Alcover_, Jul 05 2011, after _Christian G. Bower_ *)
%Y Column h=3 of A034781.
%K nonn
%O 1,5
%A _N. J. A. Sloane_
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