A005754 Number of planted identity matched trees with n nodes.
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
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
Programs
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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 *)
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
Extensions
More terms from Christian G. Bower, Dec 15 1999
Comments