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%I #10 Mar 18 2023 03:39:26
%S 1,1,12,291,10243,460632,24830853,1546531419,108716955930,
%T 8489321379453,727903248520260,67935651633100242,6853940772480079902,
%U 743261410711529857459,86224073603509482578211,10656471864208782754351131,1398062659621217619155428209
%N G.f. A(x) satisfies A(x) = Series_Reversion(x - x^3*A'(x)^3).
%H Paul D. Hanna, <a href="/A361302/b361302.txt">Table of n, a(n) for n = 1..300</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)^3).
%F (2) A(x) = x + A(x)^3 * A'(A(x))^3.
%F (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(3*n) / n! ).
%F (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(3*n) / n! is the g.f. of A361536.
%F (5) a(n) = A361536(n-1)/(2*n-1) for n >= 1.
%e G.f.: A(x) = x + x^3 + 12*x^5 + 291*x^7 + 10243*x^9 + 460632*x^11 + 24830853*x^13 + 1546531419*x^15 + 108716955930*x^17 + ... + a(n)*x^(2*n-1) + ...
%e By definition, A(x - x^3*A'(x)^3) = x, where
%e A'(x) = 1 + 3*x^2 + 60*x^4 + 2037*x^6 + 92187*x^8 + 5066952*x^10 + 322801089*x^12 + 23197971285*x^14 + ... + A361536(n)*x^(2*n) + ...
%e Also,
%e A'(x) = 1 + (d/dx x^3*A'(x)^3) + (d^2/dx^2 x^6*A'(x)^6)/2! + (d^3/dx^3 x^9*A'(x)^9)/3! + (d^4/dx^4 x^12*A'(x)^12)/4! + (d^5/dx^5 x^15*A'(x)^15)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(3*n))/n! + ...
%e Further,
%e A(x) = x * exp( x^2*A'(x)^3 + (d/dx x^5*A'(x)^6)/2! + (d^2/dx^2 x^8*A'(x)^9)/3! + (d^3/dx^3 x^11*A'(x)^12)/4! + (d^4/dx^4 x^14*A'(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(3*n))/n! + ... ).
%o (PARI) {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^3*A'^3 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A361536.
%Y Cf. A229619, A360976, A360977, A360978, A361307, A361308, A361309, A361310, A361311.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Mar 17 2023
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