Search: id:A001678
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%I A001678 M0768 N0293
%S A001678 0,0,1,0,1,1,2,3,6,10,19,35,67,127,248,482,952,1885,3765,7546,15221,30802,
%T A001678 62620,127702,261335,536278,1103600,2276499,4706985,9752585,20247033,
%U A001678 42110393,87733197,183074638,382599946,800701320,1677922740,3520581954
%N A001678 Number of series-reduced planted trees with n nodes.
%C A001678 The initial term is 0 by convention, though a good case can be made that 
               it should be 1 instead.
%D A001678 D. G. Cantor, personal communication.
%D A001678 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 525.
%D A001678 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 62.
%D A001678 F. Harary and E. M. Palmer, Probability that a point of a tree is fixed, 
               Math. Proc. Camb. Phil. Soc. 85 (1979) 407-415.
%D A001678 F. Harary and G. Prins, The number of homeomorphically irreducible trees 
               and other species, Acta Math., 101 (1959), 141-162.
%D A001678 F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for 
               determining the asymptotic number of trees of various species, J. 
               Austral. Math. Soc., Series A, 20 (1975), 483-503. Errata: Vol. A 
               41 (1986), p. 325.
%D A001678 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001678 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001678 Christian G. Bower, <a href="b001678.txt">Table of n, a(n) for n = 0..500</
               a>
%H A001678 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=404">
               Encyclopedia of Combinatorial Structures 404</a>
%H A001678 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Series-ReducedTree.html">Link to a section of The World of Mathematics.</
               a>
%H A001678 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A001678 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%F A001678 G.f. A(x) satisfies A(x) = (x^2/(1+x))*exp( sum_{k=1..infinity} A(x^k)/
               (k*x^k) ) [Harary and E. M. Palmer, 1973, p. 62, Eq. (3.3.8).
%F A001678 G.f. Sum a(n)*x^n = x^2/((1+x)*Product_{k>0}(1-x^k)^a(k+1)). - Michael 
               Somos, Oct 06 2003
%e A001678 --------------- Examples (i=internal,e=external): ---------------------------
%e A001678 |.n=2.|..n=4..|..n=5..|...n=6.............|....n=7..........................|
%e A001678 |.....|.......|.......|.............e...e.|................e.e.e......e...e.|
%e A001678 |.....|.e...e.|.e.e.e.|.e.e.e.e...e...i...|.e.e.e.e.e...e....i....e.e...i...|
%e A001678 |..e..|...i...|...i...|....i........i.....|.....i..........i..........i.....|
%e A001678 |..e..|...e...|...e...|....e........e.....|.....e..........e..........e.....|
%e A001678 -----------------------------------------------------------------------------
%p A001678 with (powseries): with (combstruct): n := 30: sys := {B = Prod(C,Z), 
               S = Set(B,1 <= card), C = Union(Z,S)}: A001678 := 1,0,1,seq(count([S, 
               sys, unlabeled],size=i),i=1..n); # from UlrSchimke(AT)aol.com
%o A001678 (PARI) a(n)=if(n<4, n==2, T(n-2,n-3)) where T(n,k)=if(n<1|k<1, (n==0)&(k>
               =0), sum(j=1,k,sum(i=1,n\j, T(n-i*j,min(n-i*j,j-1))*binomial(a(j+1)+i-1,
               i)))) /* Michael SOos Jun 04 2002 */
%o A001678 (PARI) {a(n)=local(A); if(n<3, n==2, A=x/(1-x^2)+O(x^n); for(k=3,n-2, 
               A/=(1-x^k+O(x^n))^polcoeff(A,k)); polcoeff(A,n-1))} /* Michael Somos 
               Oct 06 2003 */
%Y A001678 Cf. A000014, A007827.
%Y A001678 Sequence in context: A000693 A054178 A005833 this_sequence A113292 A050291 
               A135452
%Y A001678 Adjacent sequences: A001675 A001676 A001677 this_sequence A001679 A001680 
               A001681
%K A001678 nonn,easy,nice
%O A001678 0,7
%A A001678 N. J. A. Sloane (njas(AT)research.att.com).
%E A001678 Additional comments from Michael Somos, Jun 05, 2002.

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