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%I #42 Nov 06 2019 04:22:24
%S 1,1,3,13,69,419,2809,20353,157199,1281993,10963825,97828031,
%T 907177801,8716049417,86553001779,886573220093,9351927111901,
%U 101447092428243,1130357986741545,12923637003161409,151479552582252239
%N G.f. satisfies A(x) = 1 + x*A(x)^2*A(x*A(x)).
%C Equals row sums of triangle A291820.
%F G.f. satisfies:
%F (1) A(x) = (1/x)*Series_Reversion(x - x^2*A(x)).
%F (2) A(x) = 1 + (1/x)*Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n!.
%F (3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n/n! ).
%F (4) A(x) = 1/(1 - x*A(x)*A(x*A(x))).
%F (5) A(x) = f(x*A(x)) = (1-1/f(x))/x where f(x) is the g.f. of A088713.
%F Given g.f. A(x), then B(x) = x*A(x) satisfies 0 = f(x, B(x), B(B(x))) where f(a0, a1, a2) = a0 - a1 + a1*a2. - _Michael Somos_, May 21 2005
%F From _Paul D. Hanna_, Jul 09 2009: (Start)
%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(n+k+m,k)/(n+k+m) * a(n-k,k).
%F (End)
%F a(n) = Sum_{k=0..n} A291820(n+1,k). - _Paul D. Hanna_, Sep 01 2017
%e G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 419*x^5 + 2809*x^6 +...
%e The g.f. A(x) satisfies:
%e x*A(x) = x + x^2*A(x) + d/dx x^4*A(x)^2/2! + d^2/dx^2 x^6*A(x)^3/3! + d^3/dx^3 x^8*A(x)^4/4! +...
%e The logarithm of the g.f. is given by:
%e log(A(x)) = x*A(x) + d/dx x^3*A(x)^2/2! + d^2/dx^2 x^5*A(x)^3/3! + d^3/dx^3 x^7*A(x)^4/4! + d^4/dx^4 x^9*A(x)^5/5! +...
%e From _Paul D. Hanna_, Apr 16 2007: (Start)
%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 + x*B*C;
%e C = B + x*C*D;
%e D = C + x*D*E;
%e E = D + x*E*F ; ...
%e where variables B,C,D,E,..., are formed from successive iterations of x*A(x):
%e B = A(x)*A(x*A(x)), C = B*A(x*B), D = C*A(x*C), E = D*A(x*D), ...;
%e more explicilty,
%e B = 1 + 2*x + 8*x^2 + 42*x^3 + 258*x^4 + 1764*x^5 + 13070*x^6 +...,
%e C = 1 + 3*x + 15*x^2 + 93*x^3 + 655*x^4 + 5039*x^5 + 41453*x^6 +...,
%e D = 1 + 4*x + 24*x^2 + 172*x^3 + 1372*x^4 + 11796*x^5 +...,
%e E = 1 + 5*x + 35*x^2 + 285*x^3 + 2545*x^4 + 24255*x^5 +...,
%e ... (End)
%e Related expansions:
%e A(x*A(x)) = 1 + x + 4*x^2 + 22*x^3 + 142*x^4 + 1016*x^5 + 7838*x^6 + 64174*x^7 + 552112*x^8 +...
%e A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 173*x^4 + 1054*x^5 + 7039*x^6 + 50632*x^7 + 387613*x^8 +...
%e d/dx x^4*A(x)^2/2! = 2*x^3 + 5*x^4 + 21*x^5 + 112*x^6 + 692*x^7 + 4743*x^8 +...
%e d^2/dx^2 x^6*A(x)^3/3! = 5*x^4 + 21*x^5 + 112*x^6 + 696*x^7 + 4815*x^8 +...
%e d^3/dx^3 x^8*A(x)^4/4! = 14*x^5 + 84*x^6 + 540*x^7 + 3795*x^8 +...
%e d^4/dx^4 x^10*A(x)^5/5! = 42*x^6 + 330*x^7 + 2475*x^8 + 19305*x^9 +...
%e ...
%e d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n! = A000108(n)*x^(n+1) +...
%t m = 21; A[_] = 1; Do[A[x_] = 1 + x A[x]^2 A[x A[x]] + O[x]^m, {m}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 06 2019 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, n++; A = x + O(x^2); for(i=2, n, A = x / (1 - subst(A, x, A))); polcoeff(A, n))}; /* _Michael Somos_, May 21 2005 */
%o (PARI) {a(n)=local(A); if(n<0, 0, A=1+x+O(x^2); for(i=1,n, A=1/(1-x*A*subst(A,x,x*A)));polcoeff(A,n))}
%o (PARI) {a(n)=local(A); if(n<0, 0, A=1+x+O(x^2);for(i=0,n, A=(1/x)*serreverse(x-x^2*A));polcoeff(A,n))}
%o (PARI) {a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,m*binomial(n+k+m,k)/(n+k+m)*a(n-k,k))))} \\ _Paul D. Hanna_, Jul 09 2009
%o (PARI) /* n-th Derivative: */
%o {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o /* G.f.: [_Paul D. Hanna_, Dec 18 2010] */
%o {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n,A=exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*A^m/m!))+x*O(x^n))); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* n-th Derivative: */
%o {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o /* G.f.: [_Paul D. Hanna_, May 31 2012] */
%o {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n,A=1+(1/x)*sum(m=1, n+1, Dx(m-1, x^(2*m)*A^m/m!))+x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A291820, A088713, A212910, A212919.
%Y Apart from signs, same as A067145. - _Philippe Deléham_, Jun 18 2006
%Y Cf. A002449, A030266, A087949, A088717, A091713, A120971.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 12 2003, May 22 2008
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