A052773 A simple grammar.

1, 1, 5, 31, 229, 1832, 15583, 137791, 1255202, 11693697, 110905169, 1067181020, 10392861567, 102239342761, 1014484221699, 10141596951782, 102044286177390, 1032652191535027, 10503201188806574, 107313868098732336, 1100922685481490057, 11335843298568212815, 117111555943587032146, 1213575764038590524010

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..962
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 730

Crossrefs

Cf. A052763, A242249.

Programs

Maple

spec := [S,{S=Set(B),B=Prod(Z,S,S,S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
b:= proc(n) option remember; add(a(i)*a(n-i), i=0..n) end:
g:= proc(n) option remember; add(b(i)*b(n-i), i=0..n) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*g(d-1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 24 2017

Mathematica

b[n_] := b[n] = Sum[a[i]*a[n-i], {i, 0, n}];
g[n_] := g[n] = Sum[b[i]*b[n-i], {i, 0, n}];
a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, #*g[#-1]&]*a[n-j], {j, 1, n} ]/n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 28 2017, after Alois P. Heinz *)

PARI

{a(n)=local(A=1+x+x*O(x^n));if(n==0,1,for(i=1,n, A=exp(sum(k=1,n,subst(x*A^4,x,x^k+x*O(x^n))/k)));polcoeff(A,n,x))} \\ Paul D. Hanna, Jul 13 2006

Formula

G.f.: A(x) = exp(A(x)^4*x + A(x^2)^4*x^2/2 + A(x^3)^4*x^3/3 +...), A(0)=1; also, A(x)^4 = sum_{n=0..inf} A052763(n+1)x^n. - Paul D. Hanna, Jul 13 2006

a(n) ~ c * d^n / n^(3/2), where d = 11.069962877759326312419302623317740386289... (see d(4) in A242249, or A052763) and c = 0.131073637348549764379358468465557... . - Vaclav Kotesovec, Mar 28 2017

Extensions

More terms from Paul D. Hanna, Jul 13 2006