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A361542 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(2*n) / n!.

%I A361542 #10 Mar 17 2023 03:03:57
%S A361542 1,4,84,2940,137228,7809680,517517212,38860889496,3248881861500,
%T A361542 298704250964336,29928006672383280,3244628959712243628,
%U A361542 378449007991303855532,47261928190105905687600,6293239981401396941576632,890249832854933140207681360,133355904852469516343820132852
%N A361542 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(2*n) / n!.
%H A361542 Paul D. Hanna, <a href="/A361542/b361542.txt">Table of n, a(n) for n = 0..300</a>
%F A361542 G.f. A(x) = Sum_{n>=0} a(n)*x^(3*n) may be defined by the following.
%F A361542 (1) A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(2*n) / n!.
%F A361542 (2) A(x) = d/dx Series_Reversion(x - x^4*A(x)^2).
%F A361542 (3) B(x - x^4*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A(x)^(2*n) / n! ) is the g.f. of A361309.
%F A361542 (4) a(n) = (3*n+1) * A361309(n+1) for n >= 0.
%F A361542 a(n) ~ c * 3^n * n! * n^((10*LambertW(1/2) - 1 + 2/(1 + LambertW(1/2)))/3) / LambertW(1/2)^n, where c = 0.21905012601982153625576836... - _Vaclav Kotesovec_, Mar 16 2023
%e A361542 G.f.: A(x) = 1 + 4*x^3 + 84*x^6 + 2940*x^9 + 137228*x^12 + 7809680*x^15 + 517517212*x^18 + 38860889496*x^21 + ... + a(n)*x^(3*n) + ...
%e A361542 where
%e A361542 A(x) = 1 + (d/dx x^4*A(x)^2) + (d^2/dx^2 x^8*A(x)^4)/2! + (d^3/dx^3 x^12*A(x)^6)/3! + (d^4/dx^4 x^16*A(x)^8)/4! + (d^5/dx^5 x^20*A(x)^10)/5! + ... + (d^n/dx^n x^(4*n)*A(x)^(2*n))/n! + ...
%e A361542 Related series.
%e A361542 Let B(x) = Series_Reversion(x - x^4*A(x)^2), which begins
%e A361542 B(x) = x + x^4 + 12*x^7 + 294*x^10 + 10556*x^13 + 488105*x^16 + 27237748*x^19 + ... + A361309(n+1)*x^(3*n+1) + ...
%e A361542 then A(x) = B'(x) and
%e A361542 B(x) = x * exp( x^3*A(x)^2 + (d/dx x^7*A(x)^4)/2! + (d^2/dx^2 x^11*A(x)^6)/3! + (d^3/dx^3 x^15*A(x)^8)/4! + (d^4/dx^4 x^19*A(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A(x)^(2*n))/n! + ... ).
%t A361542 nmax = 20; r = 4; s = 2; 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 A361542 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A361542 {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(4*m)*A^(2*m)/m!)) +O(x^(3*n+1))); polcoeff(A, 3*n)}
%o A361542 for(n=0, 25, print1(a(n), ", "))
%o A361542 (PARI) /* Using series reversion (faster) */
%o A361542 {a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^4*A^2 +O(x^(3*n+2))))); polcoeff(A, 3*n)}
%o A361542 for(n=0, 25, print1(a(n), ", "))
%Y A361542 Cf. A360950, A360974, A360975, A360973, A361046, A361536, A361537, A361541, A361543, A361551.
%Y A361542 Cf. A361309.
%K A361542 nonn
%O A361542 0,2
%A A361542 _Paul D. Hanna_, Mar 15 2023