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A005754 Number of planted identity matched trees with n nodes.

%I A005754 M1765 #45 May 26 2023 08:51:05
%S A005754 1,1,2,7,24,95,388,1650,7183,31965,144502,662241,3068942,14358678,
%T A005754 67729973,321759461,1538076291,7392775328,35707198905,173221206284,
%U A005754 843634142771,4123376617009,20218897206392,99436453714990,490355165178472,2424146632435852
%N A005754 Number of planted identity matched trees with n nodes.
%C A005754 Number of rooted identity trees with n nodes and edges not attached to root are 2-colored or oriented. - _Christian G. Bower_, Dec 15 1999
%D A005754 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005754 Alois P. Heinz, <a href="/A005754/b005754.txt">Table of n, a(n) for n = 1..400</a>
%H A005754 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=430">Encyclopedia of Combinatorial Structures 430</a>
%H A005754 R. Simion, <a href="http://dx.doi.org/10.1016/0012-365X(91)90061-6">Trees with 1-factors and oriented trees</a>, Discrete Math., 88 (1991), 93-104.
%H A005754 R. Simion, <a href="/A005750/a005750.pdf">Trees with 1-factors and oriented trees</a>, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
%H A005754 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A005754 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A005754 a(n+1) is Weigh transform of A005753. - _Christian G. Bower_, Dec 15 1999
%F A005754 a(n) ~ c * d^n / n^(3/2), where d = A246312 = 5.2490324912281705791649522..., c = 0.05927840588836202377824646... . - _Vaclav Kotesovec_, Aug 25 2014
%F A005754 G.f. A(x) satisfies: A(x) = x * exp( A(x)^2/x - A(x^2)^2/(2*x^2) + A(x^3)^2/(3*x^3) - A(x^4)^2/(4*x^4) + ... ). - _Ilya Gutkovskiy_, May 26 2023
%p A005754 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A005754       add(binomial(2*b((i-1)$2), j)*b(n-i*j, i-1), j=0..n/i)))
%p A005754     end:
%p A005754 g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A005754       add(binomial(b((i-1)$2), j)*g(n-i*j, i-1), j=0..n/i)))
%p A005754     end:
%p A005754 a:= n-> g((n-1)$2):
%p A005754 seq(a(n), n=1..30);  # _Alois P. Heinz_, Aug 01 2013
%t A005754 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[2*b[i-1, i-1], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i-1, i-1], j]*g[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := g[n-1, n-1]; Table[a[n], {n, 1, 30}] // FullSimplify (* _Jean-François Alcover_, Dec 02 2013, translated from _Alois P. Heinz_'s Maple program *)
%Y A005754 Cf. A005753, A102755, A246312.
%K A005754 nonn,nice,easy
%O A005754 1,3
%A A005754 _N. J. A. Sloane_
%E A005754 More terms from _Christian G. Bower_, Dec 15 1999