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A052773 A simple grammar.

%I A052773 #23 Jul 14 2024 15:42:07
%S A052773 1,1,5,31,229,1832,15583,137791,1255202,11693697,110905169,1067181020,
%T A052773 10392861567,102239342761,1014484221699,10141596951782,
%U A052773 102044286177390,1032652191535027,10503201188806574,107313868098732336,1100922685481490057,11335843298568212815,117111555943587032146,1213575764038590524010
%N A052773 A simple grammar.
%H A052773 Alois P. Heinz, <a href="/A052773/b052773.txt">Table of n, a(n) for n = 0..962</a>
%H A052773 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=730">Encyclopedia of Combinatorial Structures 730</a>
%F A052773 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
%F A052773 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
%p A052773 spec := [S,{S=Set(B),B=Prod(Z,S,S,S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052773 # second Maple program:
%p A052773 b:= proc(n) option remember; add(a(i)*a(n-i), i=0..n) end:
%p A052773 g:= proc(n) option remember; add(b(i)*b(n-i), i=0..n) end:
%p A052773 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A052773       d*g(d-1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A052773     end:
%p A052773 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 24 2017
%t A052773 b[n_] := b[n] = Sum[a[i]*a[n-i], {i, 0, n}];
%t A052773 g[n_] := g[n] = Sum[b[i]*b[n-i], {i, 0, n}];
%t A052773 a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, #*g[#-1]&]*a[n-j], {j, 1, n} ]/n];
%t A052773 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 28 2017, after _Alois P. Heinz_ *)
%o A052773 (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
%Y A052773 Cf. A052763, A242249.
%K A052773 easy,nonn
%O A052773 0,3
%A A052773 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052773 More terms from _Paul D. Hanna_, Jul 13 2006