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%I #5 Apr 07 2022 12:12:49
%S 1,1,3,13,69,418,2785,19965,151839,1213329,10117007,87572771,
%T 783819258,7231777871,68607958358,667912243459,6661081903672,
%U 67956552527140,708354657022645,7536107197624606,81756574537050031,903706858640897865,10170700352770130132
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^(3*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)^(3*n) * A(x*A(x)^n),
%F (2) 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+3)).
%e G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 418*x^5 + 2785*x^6 + 19965*x^7 + 151839*x^8 + 1213329*x^9 + 10117007*x^10 + ...
%e where
%e (1) 1 = A(x) - x*A(x)^3*A(x*A(x)) + x^2*A(x)^6*A(x*A(x)^2) - x^3*A(x)^9*A(x*A(x)^3) + x^4*A(x)^12*A(x*A(x)^4) - x^5*A(x)^15*A(x*A(x)^5) + x^6*A(x)^18*A(x*A(x)^6) + ...
%e (2) 1 = 1/(1 + x*A(x)^3) + 1*x/(1 + x*A(x)^4) + 3*x^2/(1 + x*A(x)^5) + 13*x^3/(1 + x*A(x)^6) + 69*x^4/(1 + x*A(x)^7) + 418*x^5/(1 + x*A(x)^8) + 2785*x^6/(1 + x*A(x)^9) + ... + a(n)*x^n/(1 + x*A(x)^(n+3)) + ...
%o (PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^(3*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)^(3*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+3)) */
%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+3)) ),#A-1)); A[n+1]}
%o for(n=0,31,print1(a(n),", "))
%Y Cf. A352853, A352854, A352856.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 05 2022
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