A120971 G.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * A( x*A(x)^2 )^2.
1, 1, 4, 26, 218, 2151, 23854, 289555, 3783568, 52624689, 772928988, 11918181144, 192074926618, 3224153299106, 56213565222834, 1015694652332437, 18982833869517376, 366384235565593176, 7292660345274942402
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Keywords
Examples
G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 218*x^4 + 2151*x^5 + 23854*x^6 +... From _Paul D. Hanna_, Apr 16 2007: G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations: A = 1 + x*B^2; B = A*(1 + x*C^2); C = B*(1 + x*D^2); D = C*(1 + x*E^2); E = D*(1 + x*F^2); ... The above series begin: B(x) = 1 + 2*x + 11*x^2 + 87*x^3 + 841*x^4 + 9288*x^5 + 113166*x^6 +... C(x) = 1 + 3*x + 21*x^2 + 198*x^3 + 2204*x^4 + 27431*x^5 + 371102*x^6 +... D(x) = 1 + 4*x + 34*x^2 + 374*x^3 + 4747*x^4 + 66350*x^5 + 996943*x^6 +... E(x) = 1 + 5*x + 50*x^2 + 630*x^3 + 9015*x^4 + 140510*x^5 + 2334895*x^6 +... F(x) = 1 + 6*x + 69*x^2 + 981*x^3 + 15658*x^4 + 270016*x^5 + 4933294*x^6 +...
Crossrefs
Programs
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Mathematica
m = 19; A[] = 0; Do[A[x] = 1 + x A[x]^2 A[x A[x]^2]^2 + O[x]^m, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)
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PARI
{a(n)=local(A,G=[1,1]);for(i=1,n,G=concat(G,0); G[ #G]=-Vec(subst(Ser(G),x,x/Ser(G)^2))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/2));A[n+1]} -
PARI
a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n+k, j)/(2*n+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Mar 01 2025
Formula
G.f. A(x) satisfies:
(1) A(x) = G(G(x)-1),
(2) A(G(x)-1) = G(A(x)-1),
(3) A(x) = G(x*A(x)^2),
(4) A(x/G(x)^2) = G(x),
where G(x) is the g.f. of A120970 and satisfies G(x/G(x)^2) = 1 + x.
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+1)^2) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007
Let B(x) = Sum_{n>=0} a(n)*x^(2*n+1), then B( x/(1+B(x)^2) ) = x. - Paul D. Hanna, Oct 30 2013
From Seiichi Manyama, Mar 01 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n+k,j)/(2*n+k) * a(n-j,2*j). (End)