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G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n).
3

%I #6 Apr 07 2022 12:12:13

%S 1,1,1,2,5,15,53,205,865,3928,18943,96387,514745,2871568,16670197,

%T 100400979,625756254,4026925835,26705001158,182188059474,

%U 1276736262332,9178023547010,67597658864028,509525556949153,3926577535219879,30908466065826275,248308190295151020

%N G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n).

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

%F (1) 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n),

%F (2) 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+1)).

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 53*x^6 + 205*x^7 + 865*x^8 + 3928*x^9 + 18943*x^10 + 96387*x^11 + 514745*x^12 + ...

%e where

%e (1) 1 = A(x) - x*A(x)*A(x*A(x)) + x^2*A(x)^2*A(x*A(x)^2) - x^3*A(x)^3*A(x*A(x)^3) + x^4*A(x)^4*A(x*A(x)^4) - x^5*A(x)^5*A(x*A(x)^5) + x^6*A(x)^6*A(x*A(x)^6) + ...

%e (2) 1 = 1/(1 + x*A(x)) + 1*x/(1 + x*A(x)^2) + 1*x^2/(1 + x*A(x)^3) + 2*x^3/(1 + x*A(x)^4) + 5*x^4/(1 + x*A(x)^5) + 15*x^5/(1 + x*A(x)^6) + 53*x^6/(1 + x*A(x)^7) + ... + a(n)*x^n/(1 + x*A(x)^(n+1)) + ...

%o (PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n) */

%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);

%o A[#A] = -polcoeff( sum(n=0,#A-1, (-x)^n*Ser(A)^n*subst(Ser(A),x,x*Ser(A)^n) ),#A-1)); A[n+1]}

%o for(n=0,31,print1(a(n),", "))

%o (PARI) /* 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+1)) */

%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);

%o A[#A] = -polcoeff( sum(n=0,#A-1, A[n+1]*x^n/(1 + x*Ser(A)^(n+1)) ),#A-1)); A[n+1]}

%o for(n=0,31,print1(a(n),", "))

%Y Cf. A352854, A352855, A352856.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Apr 05 2022