A052797 Number of rooted identity trees with n nodes and 5-colored non-root nodes.

0, 1, 5, 35, 310, 2980, 30526, 325655, 3581200, 40301850, 461925625, 5373527605, 63281397830, 752962948360, 9038406399150, 109321688626100, 1331047556550240, 16300720361555725, 200658900798443135, 2481460407963908510, 30814094488256704650, 384069865485447909905

Offset

0,3

Comments

Previous name was: A simple grammar.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 754

Formula

a(n) ~ c * d^n / n^(3/2), where d = 13.40708747253774757978704707270263863994591470583713..., c = 0.07868285364894808762720738672045999... . - Vaclav Kotesovec, Feb 24 2015

From Ilya Gutkovskiy, Apr 13 2019: (Start)

G.f. A(x) satisfies: A(x) = x*exp(5*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(5*a(n)). (End)

Maple

spec := [S, {B=PowerSet(S), S=Prod(Z, B, B, B, B, B)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(5*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=0..25);  # Alois P. Heinz, Feb 24 2015

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[5*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], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

Crossrefs

Column k=5 of A255517.

Sequence in context: A305964 A226739 A109253 * A371540 A225177 A151344

Adjacent sequences: A052794 A052795 A052796 * A052798 A052799 A052800

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Extensions

New name from Vaclav Kotesovec, Feb 24 2015

Status

approved