This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A005753 M1514 #67 Dec 26 2020 17:31:40
%S A005753 1,2,5,18,66,266,1111,4792,21124,94888,432415,1994828,9296712,
%T A005753 43706722,207030398,987130456,4733961435,22819241034,110500644857,
%U A005753 537295738556,2622248720234,12840953621208,63074566121245,310693364823376,1534374047239554,7595642577152762
%N A005753 Number of rooted identity matched trees with n nodes.
%C A005753 Also number of rooted identity trees with n nodes and 2-colored non-root nodes. - _Christian G. Bower_, Apr 15 1998
%D A005753 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005753 Alois P. Heinz, <a href="/A005753/b005753.txt">Table of n, a(n) for n = 1..1000</a>
%H A005753 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=429">Encyclopedia of Combinatorial Structures 429</a>
%H A005753 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 A005753 R. Simion, <a href="/A005750/a005750.pdf">Trees with 1-factors and oriented trees</a>, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
%H A005753 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A005753 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A005753 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 A005753 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 A005753 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 A005753 G.f.: A(x) = x + 2*x^2 + 5*x^3 + 18*x^4 + 66*x^5 + 266*x^6 + ...
%e A005753 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 A005753 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A005753 add(binomial(2*a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A005753 end:
%p A005753 a:= n-> `if`(n=1, 1, b((n-1)$2)):
%p A005753 seq(a(n), n=1..40); # _Alois P. Heinz_, Aug 01 2013
%t A005753 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 A005753 (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 A005753 Cf. A038077, A246312.
%Y A005753 Column k=2 of A255517.
%K A005753 nonn,eigen
%O A005753 1,2
%A A005753 _N. J. A. Sloane_