%I A001383 M1107 N0422
%S A001383 1,1,1,2,4,8,15,29,53,98,177,319,565,1001,1749,3047,5264,9054,15467,
%T A001383 26320,44532,75054,125904,210413,350215,580901,960035,1581534,2596913,
%U A001383 4251486,6939635,11296231,18337815,29692431,47956995,77271074,124212966
%N A001383 Number of n-node rooted trees of height at most 3.
%D A001383 J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. 
               Dev. 4 (1960), 473-478.
%D A001383 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001383 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001383 N. J. A. Sloane, <a href="b001383.txt">Table of n, a(n) for n=0..200</
               a>
%H A001383 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=62">
               Encyclopedia of Combinatorial Structures 62</a>
%H A001383 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A001383 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%F A001383 G.f.: S[ 3 ] := x*Product (1 - x^k)^(-p(k-1)), where p(k) = number of 
               partitions of k.
%F A001383 a(n+1) is the Euler transform of p(n-1), where p() = A000041 is the partition 
               function. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 01 
               2006
%p A001383 s[ 2 ] := x/product('1-x^i','i'=1..30); # G.f. for trees of ht <=2, A000041
%p A001383 for k from 3 to 12 do # gets g.f. for trees of ht <= 3,4,5,...
%p A001383 s[ k ] := series(x/product('(1-x^i)^coeff(s[ k-1 ],x,i)','i'=1..30),x,
               31); od:
%p A001383 For Maple program see link in A000235.
%p A001383 with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; 
               local d,j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), 
               j=1..n)/n fi end end: A000041:= etr (n-> 1): a:= n->`if`(n=0,1, etr 
               (k-> A000041(k-1))(n-1)): seq (a(n), n=0..36); [From Alois P. Heinz 
               (heinz(AT)hs-heilbronn.de), Sep 08 2008]
%Y A001383 Cf. A000041, A001383-A001385, A034823-A034826.
%Y A001383 Sequence in context: A118870 A088532 A036621 this_sequence A108564 A066369 
               A000078
%Y A001383 Adjacent sequences: A001380 A001381 A001382 this_sequence A001384 A001385 
               A001386
%K A001383 nonn,easy,nice
%O A001383 0,4
%A A001383 N. J. A. Sloane (njas(AT)research.att.com).