cp's OEIS Frontend

A120971 G.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * A( x*A(x)^2 )^2.

%I A120971 #22 Mar 01 2025 08:36:03
%S A120971 1,1,4,26,218,2151,23854,289555,3783568,52624689,772928988,
%T A120971 11918181144,192074926618,3224153299106,56213565222834,
%U A120971 1015694652332437,18982833869517376,366384235565593176,7292660345274942402
%N A120971 G.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * A( x*A(x)^2 )^2.
%F A120971 G.f. A(x) satisfies:
%F A120971 (1) A(x) = G(G(x)-1),
%F A120971 (2) A(G(x)-1) = G(A(x)-1),
%F A120971 (3) A(x) = G(x*A(x)^2),
%F A120971 (4) A(x/G(x)^2) = G(x),
%F A120971 where G(x) is the g.f. of A120970 and satisfies G(x/G(x)^2) = 1 + x.
%F A120971 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
%F A120971 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
%F A120971 From _Seiichi Manyama_, Mar 01 2025: (Start)
%F A120971 Let a(n,k) = [x^n] A(x)^k.
%F A120971 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)
%e A120971 G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 218*x^4 + 2151*x^5 + 23854*x^6 +...
%e A120971 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:
%e A120971 A = 1 + x*B^2;
%e A120971 B = A*(1 + x*C^2);
%e A120971 C = B*(1 + x*D^2);
%e A120971 D = C*(1 + x*E^2);
%e A120971 E = D*(1 + x*F^2); ...
%e A120971 The above series begin:
%e A120971 B(x) = 1 + 2*x + 11*x^2 + 87*x^3 + 841*x^4 + 9288*x^5 + 113166*x^6 +...
%e A120971 C(x) = 1 + 3*x + 21*x^2 + 198*x^3 + 2204*x^4 + 27431*x^5 + 371102*x^6 +...
%e A120971 D(x) = 1 + 4*x + 34*x^2 + 374*x^3 + 4747*x^4 + 66350*x^5 + 996943*x^6 +...
%e A120971 E(x) = 1 + 5*x + 50*x^2 + 630*x^3 + 9015*x^4 + 140510*x^5 + 2334895*x^6 +...
%e A120971 F(x) = 1 + 6*x + 69*x^2 + 981*x^3 + 15658*x^4 + 270016*x^5 + 4933294*x^6 +...
%t A120971 m = 19; A[_] = 0; Do[A[x_] = 1 + x A[x]^2 A[x A[x]^2]^2 + O[x]^m, {m}];
%t A120971 CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 07 2019 *)
%o A120971 (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]}
%o A120971 (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
%Y A120971 Cf. A120970; variants: A110447, A120973, A120975, A120977.
%Y A120971 Cf. A002449, A087949, A088714, A088717, A091713.
%K A120971 nonn
%O A120971 0,3
%A A120971 _Paul D. Hanna_, Jul 20 2006