A088717 G.f. satisfies: A(x) = 1 + x*A(x)^2*A(x*A(x)^2).
1, 1, 3, 14, 84, 596, 4785, 42349, 406287, 4176971, 45640572, 526788153, 6392402793, 81247489335, 1078331283648, 14907041720241, 214187010762831, 3192620516380376, 49287883925072010, 786925082232918304, 12976244331714379149, 220728563512663520510
Offset: 0
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 596*x^5 + 4785*x^6 +... G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations: A = 1 + x*A*B; B = A*(1 + x*B*C); C = B*(1 + x*C*D); D = C*(1 + x*D*E); E = D*(1 + x*E*F); ... where B(x) = A(x)*A(x*A(x)^2), C(x) = A(x)*B(x*A(x)^2), D(x) = A(x)*C(x*A(x)^2), ... Expansions of a few of the functions described above begin: B(x) = 1 + 2*x + 9*x^2 + 55*x^3 + 402*x^4 + 3328*x^5 + 30312*x^6 +... C(x) = 1 + 3*x + 18*x^2 + 138*x^3 + 1218*x^4 + 11856*x^5 + 124467*x^6 +... D(x) = 1 + 4*x + 30*x^2 + 278*x^3 + 2901*x^4 + 32846*x^5 + 395913*x^6 +... ALTERNATE GENERATING METHOD. Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy: A = 1 + x*A^2*B, B = 1 + x*(A*B)^2*C, C = 1 + x*(A*B*C)^2*D, D = 1 + x*(A*B*C*D)^2*E, etc., then B(x) = A(x*A(x)^2), C(x) = B(x*A(x)^2), D(x) = C(x*A(x)^2), etc., where A(x) = 1 + x*A(x)^2*A(x*A(x)^2) is the g.f. of this sequence. Expansions of a few of the functions described above begin: B(x) = 1 + x + 5*x^2 + 33*x^3 + 256*x^4 + 2223*x^5 + 21058*x^6 +... C(x) = 1 + x + 7*x^2 + 60*x^3 + 578*x^4 + 6045*x^5 + 67421*x^6 +... D(x) = 1 + x + 9*x^2 + 95*x^3 + 1098*x^4 + 13526*x^5 + 175176*x^6 +...
Crossrefs
Programs
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Mathematica
m = 22; A[] = 0; Do[A[x] = 1 + x A[x]^2 A[x A[x]^2] + O[x]^m, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)
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PARI
{a(n)=local(A=1+x);for(i=0,n,A=1+x*A^2*subst(A,x,x*A^2+x*O(x^n)));polcoeff(A,n)} -
PARI
/* a(n) = [x^n] (1+x*A(x))^(2*n+1)/(2*n+1): */ {a(n)=local(A=1+x); for(i=0, n, A=sum(m=0,n,polcoeff((1+x*A+x*O(x^m))^(2*m+1)/(2*m+1),m)*x^m)+x*O(x^n));polcoeff(A,n)} -
PARI
{a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(2*n+m, k)/(2*n+m)*a(n-k, k))))}
Formula
a(n) = coefficient of x^n in (1+x*A(x))^(2*n+1)/(2*n+1) where A(x) = Sum_{n=0} a(n)*x^n.
Recurrence:
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} m*C(2n+m,k)/(2n+m) * a(n-k,k). [Paul D. Hanna, Dec 16 2010]
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n)*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007