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%I #10 Mar 16 2023 04:27:03
%S 1,3,75,3234,186471,13063908,1060481214,97053553710,9840717984447,
%T 1092337371705273,131589391554509112,17089208887923714204,
%U 2379797411747290723350,353790840030976298935989,55935780589531899802966062,9373903063348266793396858620
%N Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(4*n) / n!.
%H Paul D. Hanna, <a href="/A361537/b361537.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)^(4*n) / n!.
%F (2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^4).
%F (3) B(x - x^3*A(x)^4) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(4*n) / n! ) is the g.f. of A361307.
%F (4) a(n) = (2*n+1) * A361307(n+1) for n >= 0.
%F a(n) ~ c * 2^n * n! * n^((27*LambertW(1/4) - 1 + 3/(1 + LambertW(1/4)))/4) / LambertW(1/4)^n, where c = 0.09189708135429625612601629... - _Vaclav Kotesovec_, Mar 16 2023
%e G.f.: A(x) = 1 + 3*x^2 + 75*x^4 + 3234*x^6 + 186471*x^8 + 13063908*x^10 + 1060481214*x^12 + 97053553710*x^14 + ... + a(n)*x^(2*n) + ...
%e where
%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 Related series.
%e Let B(x) = Series_Reversion(x - x^3*A(x)^4), which begins
%e B(x) = x + x^3 + 15*x^5 + 462*x^7 + 20719*x^9 + 1187628*x^11 + 81575478*x^13 + ... + A361307(n+1)*x^(2*n+1) + ...
%e then A(x) = B'(x) and
%e B(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! + ... ).
%t nmax = 20; r = 3; s = 4; 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^(4*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^4 +O(x^(2*n+2))))); polcoeff(A, 2*n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A360950, A360974, A360975, A360973, A361046, A361536, A361541, A361542, A361543, A361551.
%Y Cf. A361307.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 15 2023
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