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%I #10 Mar 16 2023 04:26:54
%S 1,3,60,2037,92187,5066952,322801089,23197971285,1848188250810,
%T 161297106209607,15285968218925460,1562519987561305566,
%U 171348519312001997550,20068058089211306151393,2500498134501774994768119,330350627790472265384885061,46136067767500181432129130897
%N Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(3*n) / n!.
%H Paul D. Hanna, <a href="/A361536/b361536.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
%F (1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(3*n) / n!.
%F (2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^3).
%F (3) B(x - x^3*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(3*n) / n! ) is the g.f. of A361302.
%F (4) a(n) = (2*n+1) * A361302(n+1) for n >= 0.
%F a(n) ~ c * 2^n * n! * n^((21*LambertW(1/3) - 1 + 3/(1 + LambertW(1/3)))/4) / LambertW(1/3)^n, where c = 0.123530460429388663183565497... - _Vaclav Kotesovec_, Mar 16 2023
%e G.f.: A(x) = 1 + 3*x^2 + 60*x^4 + 2037*x^6 + 92187*x^8 + 5066952*x^10 + 322801089*x^12 + 23197971285*x^14 + ... + a(n)*x^(2*n) + ...
%e where
%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 Related series.
%e Let B(x) = Series_Reversion(x - x^3*A(x)^3), which begins
%e B(x) = x + x^3 + 12*x^5 + 291*x^7 + 10243*x^9 + 460632*x^11 + 24830853*x^13 + ... + A361302(n+1)*x^(2*n+1) + ...
%e then A(x) = B'(x) and
%e B(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! + ... ).
%t nmax = 20; r = 3; s = 3; A[_] = 0; Do[A[x_] = D[Normal[InverseSeries[x - x^r*A[x]^s + O[x]^k]], x], {k, 1, (r-1)*(nmax+1)+r}]; Table[CoefficientList[A[x], x][[j]], {j, 1, (r-1)*(nmax+1), r-1}] (* _Vaclav Kotesovec_, Mar 16 2023 *)
%o (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^(3*m)/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* Using series reversion (faster) */
%o {a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A^3 +O(x^(2*n+2))))); polcoeff(A, 2*n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A360950, A360974, A360975, A360973, A361046, A361537, A361541, A361542, A361543, A361551.
%Y Cf. A361302.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 15 2023
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