A052797 - OEIS

%I #30 Apr 13 2019 22:19:37

%S 0,1,5,35,310,2980,30526,325655,3581200,40301850,461925625,5373527605,

%T 63281397830,752962948360,9038406399150,109321688626100,

%U 1331047556550240,16300720361555725,200658900798443135,2481460407963908510,30814094488256704650,384069865485447909905

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

%C Previous name was: A simple grammar.

%H Alois P. Heinz, <a href="/A052797/b052797.txt">Table of n, a(n) for n = 0..500</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=754">Encyclopedia of Combinatorial Structures 754</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 13.40708747253774757978704707270263863994591470583713..., c = 0.07868285364894808762720738672045999... . - _Vaclav Kotesovec_, Feb 24 2015

%F From _Ilya Gutkovskiy_, Apr 13 2019: (Start)

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

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

%p spec := [S,{B=PowerSet(S),S=Prod(Z,B,B,B,B,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(binomial(5*a(i), j)*b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p a:= n-> `if`(n=1, 1, b((n-1)$2)):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Feb 24 2015

%t 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_ *)

%Y Column k=5 of A255517.

%K easy,nonn

%O 0,3

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

%E New name from _Vaclav Kotesovec_, Feb 24 2015