A005753 - OEIS

%I M1514 #67 Dec 26 2020 17:31:40

%S 1,2,5,18,66,266,1111,4792,21124,94888,432415,1994828,9296712,

%T 43706722,207030398,987130456,4733961435,22819241034,110500644857,

%U 537295738556,2622248720234,12840953621208,63074566121245,310693364823376,1534374047239554,7595642577152762

%N Number of rooted identity matched trees with n nodes.

%C Also number of rooted identity trees with n nodes and 2-colored non-root nodes. - _Christian G. Bower_, Apr 15 1998

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A005753/b005753.txt">Table of n, a(n) for n = 1..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=429">Encyclopedia of Combinatorial Structures 429</a>

%H 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 R. Simion, <a href="/A005750/a005750.pdf">Trees with 1-factors and oriented trees</a>, Discrete Math., 88 (1981), 97. (Annotated scanned copy)

%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 G.f.: x*Product_{n>=1} (1 + x^n)^(2*a(n)) = Sum_{n>=1} a(n)*x^n. - _Paul D. Hanna_, Dec 31 2011

%F a(n) ~ c * d^n / n^(3/2), where d = A246312 = 5.249032491228170579164952216..., c = 0.192066288645200371237879149260484794708740197522264442948290580404909605849... - _Vaclav Kotesovec_, Aug 25 2014, updated Dec 26 2020

%F G.f. A(x) satisfies: A(x) = x*exp(2*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k). - _Ilya Gutkovskiy_, Apr 13 2019

%e G.f.: A(x) = x + 2*x^2 + 5*x^3 + 18*x^4 + 66*x^5 + 266*x^6 + ...

%e where A(x) = x*(1+x)^2*(1+x^2)^4*(1+x^3)^10*(1+x^4)^36*(1+x^5)^132*... (the exponents are A038077(n), n>=1).

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(binomial(2*a(i), j)*b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p a:= n-> `if`(n=1, 1, b((n-1)$2)):

%p seq(a(n), n=1..40);  # _Alois P. Heinz_, Aug 01 2013

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[2*a[i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n == 1, 1, b[n-1, n-1]]; Table[a[n] // FullSimplify, {n, 1, 30}] (* _Jean-François Alcover_, Mar 17 2014, after _Alois P. Heinz_ *)

%o (PARI) {a(n)=polcoeff(x*prod(k=1, n-1, (1+x^k+x*O(x^n))^(2*a(k))), n)} /* _Paul D. Hanna_ */

%Y Cf. A038077, A246312.

%Y Column k=2 of A255517.

%K nonn,eigen

%O 1,2

%A _N. J. A. Sloane_