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A361541 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^n / n!.

%I A361541 #10 Mar 16 2023 04:27:14
%S A361541 1,4,56,1220,34788,1203152,48418384,2210163032,112501779300,
%T A361541 6308565897088,386149471644704,25614932030415636,1830512170952711968,
%U A361541 140224558208217547440,11464991752291729651224,996723500374559386157920,91824970792933898453830680
%N A361541 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^n / n!.
%H A361541 Paul D. Hanna, <a href="/A361541/b361541.txt">Table of n, a(n) for n = 0..300</a>
%F A361541 G.f. A(x) = Sum_{n>=0} a(n)*x^(3*n) may be defined by the following.
%F A361541 (1) A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^n / n!.
%F A361541 (2) A(x) = d/dx Series_Reversion(x - x^4*A(x)).
%F A361541 (3) B(x - x^4*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A(x)^n / n! ) is the g.f. of A361308.
%F A361541 (4) a(n) = (3*n+1) * A361308(n+1) for n >= 0.
%F A361541 a(n) ~ c * 3^n * n! * n^((6*LambertW(1) - 1 + 2/(1 + LambertW(1)))/3) / LambertW(1)^n, where c = 0.4150199735492722539427... - _Vaclav Kotesovec_, Mar 16 2023
%e A361541 G.f.: A(x) = 1 + 4*x^3 + 56*x^6 + 1220*x^9 + 34788*x^12 + 1203152*x^15 + 48418384*x^18 + 2210163032*x^21 + ... + a(n)*x^(3*n) + ...
%e A361541 where
%e A361541 A(x) = 1 + (d/dx x^4*A(x)) + (d^2/dx^2 x^8*A(x)^2)/2! + (d^3/dx^3 x^12*A(x)^3)/3! + (d^4/dx^4 x^16*A(x)^4)/4! + (d^5/dx^5 x^20*A(x)^5)/5! + ... + (d^n/dx^n x^(4*n)*A(x)^n)/n! + ...
%e A361541 Related series.
%e A361541 Let B(x) = Series_Reversion(x - x^4*A(x)), which begins
%e A361541 B(x) = x + x^4 + 8*x^7 + 122*x^10 + 2676*x^13 + 75197*x^16 + 2548336*x^19 + ... + A361308(n+1)*x^(3*n+1) + ...
%e A361541 then A(x) = B'(x) and
%e A361541 B(x) = x * exp( x^3*A(x) + (d/dx x^7*A(x)^2)/2! + (d^2/dx^2 x^11*A(x)^3)/3! + (d^3/dx^3 x^15*A(x)^4)/4! + (d^4/dx^4 x^19*A(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A(x)^n)/n! + ... ).
%t A361541 nmax = 20; r = 4; s = 1; 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 A361541 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A361541 {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(4*m)*A^(1*m)/m!)) +O(x^(3*n+1))); polcoeff(A, 3*n)}
%o A361541 for(n=0, 25, print1(a(n), ", "))
%o A361541 (PARI) /* Using series reversion (faster) */
%o A361541 {a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^4*A^1 +O(x^(3*n+2))))); polcoeff(A, 3*n)}
%o A361541 for(n=0, 25, print1(a(n), ", "))
%Y A361541 Cf. A360950, A360974, A360975, A360973, A361046, A361536, A361537, A361542, A361543, A361551.
%Y A361541 Cf. A361308.
%K A361541 nonn
%O A361541 0,2
%A A361541 _Paul D. Hanna_, Mar 15 2023