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%I #9 Mar 18 2023 03:39:59
%S 1,1,15,462,20719,1187628,81575478,6470236914,578865763791,
%T 57491440616067,6266161502595672,743009082083639748,
%U 95191896469891628934,13103364445591714775407,1928820020328686200102278,302383969785427961077318020,50307405653295945234562827135
%N G.f. A(x) satisfies A(x) = Series_Reversion(x - x^3*A'(x)^4).
%H Paul D. Hanna, <a href="/A361307/b361307.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)^4).
%F (2) A(x) = x + A(x)^3 * A'(A(x))^4.
%F (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(4*n) / n! ).
%F (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(4*n) / n! is the g.f. of A361537.
%F (5) a(n) = A361537(n-1)/(2*n-1) for n >= 1.
%e G.f.: A(x) = x + x^3 + 15*x^5 + 462*x^7 + 20719*x^9 + 1187628*x^11 + 81575478*x^13 + 6470236914*x^15 + 578865763791*x^17 + ... + a(n)*x^(2*n-1) + ...
%e By definition, A(x - x^3*A'(x)^4) = x, where
%e A'(x) = 1 + 3*x^2 + 75*x^4 + 3234*x^6 + 186471*x^8 + 13063908*x^10 + 1060481214*x^12 + 97053553710*x^14 + ... + A361537(n)*x^(2*n) + ...
%e Also,
%e A'(x) = 1 + (d/dx x^3*A'(x)^4) + (d^2/dx^2 x^6*A'(x)^8)/2! + (d^3/dx^3 x^9*A'(x)^12)/3! + (d^4/dx^4 x^12*A'(x)^16)/4! + (d^5/dx^5 x^15*A'(x)^20)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(4*n))/n! + ...
%e Further,
%e A(x) = x * exp( x^2*A'(x)^4 + (d/dx x^5*A'(x)^8)/2! + (d^2/dx^2 x^8*A'(x)^12)/3! + (d^3/dx^3 x^11*A'(x)^16)/4! + (d^4/dx^4 x^14*A'(x)^20)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(4*n))/n! + ... ).
%o (PARI) {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^3*A'^4 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A361537.
%Y Cf. A229619, A360976, A360977, A360978, A361302, A361308, A361309, A361310, A361311.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Mar 17 2023
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