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%I #31 Nov 07 2019 05:25:33
%S 1,1,3,14,84,596,4785,42349,406287,4176971,45640572,526788153,
%T 6392402793,81247489335,1078331283648,14907041720241,214187010762831,
%U 3192620516380376,49287883925072010,786925082232918304,12976244331714379149,220728563512663520510
%N G.f. satisfies: A(x) = 1 + x*A(x)^2*A(x*A(x)^2).
%F 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.
%F Recurrence:
%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
%F a(n,m) = Sum_{k=0..n} m*C(2n+m,k)/(2n+m) * a(n-k,k). [_Paul D. Hanna_, Dec 16 2010]
%F 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
%e G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 596*x^5 + 4785*x^6 +...
%e G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
%e A = 1 + x*A*B;
%e B = A*(1 + x*B*C);
%e C = B*(1 + x*C*D);
%e D = C*(1 + x*D*E);
%e E = D*(1 + x*E*F); ...
%e 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), ...
%e Expansions of a few of the functions described above begin:
%e B(x) = 1 + 2*x + 9*x^2 + 55*x^3 + 402*x^4 + 3328*x^5 + 30312*x^6 +...
%e C(x) = 1 + 3*x + 18*x^2 + 138*x^3 + 1218*x^4 + 11856*x^5 + 124467*x^6 +...
%e D(x) = 1 + 4*x + 30*x^2 + 278*x^3 + 2901*x^4 + 32846*x^5 + 395913*x^6 +...
%e ALTERNATE GENERATING METHOD.
%e Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy:
%e A = 1 + x*A^2*B,
%e B = 1 + x*(A*B)^2*C,
%e C = 1 + x*(A*B*C)^2*D,
%e D = 1 + x*(A*B*C*D)^2*E, etc.,
%e then B(x) = A(x*A(x)^2), C(x) = B(x*A(x)^2), D(x) = C(x*A(x)^2), etc.,
%e where A(x) = 1 + x*A(x)^2*A(x*A(x)^2) is the g.f. of this sequence.
%e Expansions of a few of the functions described above begin:
%e B(x) = 1 + x + 5*x^2 + 33*x^3 + 256*x^4 + 2223*x^5 + 21058*x^6 +...
%e C(x) = 1 + x + 7*x^2 + 60*x^3 + 578*x^4 + 6045*x^5 + 67421*x^6 +...
%e D(x) = 1 + x + 9*x^2 + 95*x^3 + 1098*x^4 + 13526*x^5 + 175176*x^6 +...
%t m = 22; A[_] = 0; Do[A[x_] = 1 + x A[x]^2 A[x A[x]^2] + O[x]^m, {m}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 07 2019 *)
%o (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)}
%o (PARI) /* a(n) = [x^n] (1+x*A(x))^(2*n+1)/(2*n+1): */
%o {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)}
%o (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))))}
%Y Cf. A215505, A215506, A215507.
%Y Cf. A182224, A002449, A030266, A087949, A088714, A091713, A120971.
%K nonn,eigen
%O 0,3
%A _Paul D. Hanna_, Oct 12 2003 and Mar 10 2007
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