A005754 Number of planted identity matched trees with n nodes. (Formerly M1765)

1, 1, 2, 7, 24, 95, 388, 1650, 7183, 31965, 144502, 662241, 3068942, 14358678, 67729973, 321759461, 1538076291, 7392775328, 35707198905, 173221206284, 843634142771, 4123376617009, 20218897206392, 99436453714990, 490355165178472, 2424146632435852

Offset

1,3

Comments

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

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..400

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 430

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)

N. J. A. Sloane, Transforms

Index entries for sequences related to rooted trees

Formula

a(n+1) is Weigh transform of A005753. - Christian G. Bower, Dec 15 1999

a(n) ~ c * d^n / n^(3/2), where d = A246312 = 5.2490324912281705791649522..., c = 0.05927840588836202377824646... . - Vaclav Kotesovec, Aug 25 2014

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

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2*b((i-1)$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(b((i-1)$2), j)*g(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> g((n-1)$2):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 01 2013

Mathematica

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 *)

Crossrefs

Cf. A005753, A102755, A246312.

Sequence in context: A150421 A150422 A137952 * A007162 A150423 A150424

Adjacent sequences: A005751 A005752 A005753 * A005755 A005756 A005757

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower, Dec 15 1999

Status

approved