A005753 Number of rooted identity matched trees with n nodes. (Formerly M1514)

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

1,2

Comments

Also number of rooted identity trees with n nodes and 2-colored non-root nodes. - Christian G. Bower, Apr 15 1998

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

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

Example

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).

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 *)

Prog

(PARI) {a(n)=polcoeff(x*prod(k=1, n-1, (1+x^k+x*O(x^n))^(2*a(k))), n)} /* Paul D. Hanna */

Crossrefs

Cf. A038077, A246312.

Column k=2 of A255517.

Sequence in context: A150017 A150018 A150019 * A150020 A144721 A150021

Adjacent sequences: A005750 A005751 A005752 * A005754 A005755 A005756

Keyword

nonn,eigen

Author

N. J. A. Sloane

Status

approved