A005753 Number of rooted identity matched trees with n nodes.
1, 2, 5, 18, 66, 266, 1111, 4792, 21124, 94888, 432415, 1994828, 9296712, 43706722, 207030398, 987130456, 4733961435, 22819241034, 110500644857, 537295738556, 2622248720234, 12840953621208, 63074566121245, 310693364823376, 1534374047239554, 7595642577152762
Offset: 1
Examples
G.f.: A(x) = x + 2*x^2 + 5*x^3 + 18*x^4 + 66*x^5 + 266*x^6 + ... 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).
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 429
- R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1991), 93-104.
- R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
- Index entries for sequences related to rooted trees
- Index entries for sequences related to trees
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(2*a(i), j)*b(n-i*j, i-1), j=0..n/i))) end: a:= n-> `if`(n=1, 1, b((n-1)$2)): seq(a(n), n=1..40); # Alois P. Heinz, Aug 01 2013 -
Mathematica
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 *) -
PARI
{a(n)=polcoeff(x*prod(k=1, n-1, (1+x^k+x*O(x^n))^(2*a(k))), n)} /* Paul D. Hanna */
Formula
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
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
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
Comments