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G.f. satisfies: A(x) = Series_Reversion(x - x^3*A'(x)).
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%I #14 Feb 28 2023 04:26:06

%S 1,1,6,66,1027,20274,479403,13118703,406181493,14007078204,

%T 531778565544,22028404578840,988535991793203,47773653611710429,

%U 2473958531200630992,136684964338470273828,8026375457238402039978,499251236257852169668461,32794618460003080060574283

%N G.f. satisfies: A(x) = Series_Reversion(x - x^3*A'(x)).

%C a(n) = A360973(n-1)/(2*n-1) for n >= 1.

%H Paul D. Hanna, <a href="/A360976/b360976.txt">Table of n, a(n) for n = 1..200</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.

%F (1) A(x) = Series_Reversion(x - x^3*A'(x)).

%F (2) A(x) = x + A(x)^3 * A'(A(x)).

%F (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^n / n! ).

%F (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^n / n!.

%e G.f.: A(x) = x + x^3 + 6*x^5 + 66*x^7 + 1027*x^9 + 20274*x^11 + 479403*x^13 + 13118703*x^15 + 406181493*x^17 + ... + a(n)*x^(2*n-1) + ...

%e By definition, A(x - x^3*A'(x)) = x, where

%e A'(x) = 1 + 3*x^2 + 30*x^4 + 462*x^6 + 9243*x^8 + 223014*x^10 + 6232239*x^12 + 196780545*x^14 + ... + A360973(n)*x^(2*n) + ...

%e Also,

%e A'(x) = 1 + (d/dx x^3*A'(x)) + (d^2/dx^2 x^6*A'(x)^2)/2! + (d^3/dx^3 x^9*A'(x)^3)/3! + (d^4/dx^4 x^12*A'(x)^4)/4! + (d^5/dx^5 x^15*A'(x)^5)/5! + (d^6/dx^6 x^18*A'(x)^6)/6! + ... + (d^n/dx^n x^(3*n)*A'(x)^n)/n! + ...

%e Further,

%e A(x) = x * exp( x^2*A'(x) + (d/dx x^5*A'(x)^2)/2! + (d^2/dx^2 x^8*A'(x)^3)/3! + (d^3/dx^3 x^11*A'(x)^4)/4! + (d^4/dx^4 x^14*A'(x)^5)/5! + (d^5/dx^5 x^17*A'(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^n)/n! + ... ).

%o (PARI) {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^3*A'+x*O(x^(2*n)))); polcoeff(A, 2*n-1)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A360973, A229619, A360977, A360978.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Feb 27 2023