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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..962

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 730

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

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 *)

Prog

(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

Crossrefs

Cf. A052763, A242249.

Sequence in context: A365180 A192950 A001910 * A062147 A213048 A349535

Adjacent sequences: A052770 A052771 A052772 * A052774 A052775 A052776

Keyword

easy,nonn

Author

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

Extensions

More terms from Paul D. Hanna, Jul 13 2006

Status

approved