cp's OEIS Frontend

A361543 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(3*n) / n!.

%I A361543 #9 Mar 17 2023 03:00:59
%S A361543 1,4,112,5380,346788,27285968,2498963752,259124694312,29885849525700,
%T A361543 3786931724896768,522451837498888672,77929657518224116484,
%U A361543 12496899169394954817144,2144326582901160246138160,392104633203721656029928184,76134826269461672101153285664
%N A361543 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(3*n) / n!.
%H A361543 Paul D. Hanna, <a href="/A361543/b361543.txt">Table of n, a(n) for n = 0..300</a>
%F A361543 G.f. A(x) = Sum_{n>=0} a(n)*x^(3*n) may be defined by the following.
%F A361543 (1) A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(3*n) / n!.
%F A361543 (2) A(x) = d/dx Series_Reversion(x - x^4*A(x)^3).
%F A361543 (3) B(x - x^4*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A(x)^(3*n) / n! ) is the g.f. of A361310.
%F A361543 (4) a(n) = (3*n+1) * A361310(n+1) for n >= 0.
%F A361543 a(n) ~ c * 3^n * n! * n^((14*LambertW(1/3) - 1 + 2/(1 + LambertW(1/3)))/3) / LambertW(1/3)^n, where c = 0.147646967535758066931690294... - _Vaclav Kotesovec_, Mar 16 2023
%e A361543 G.f.: A(x) = 1 + 4*x^3 + 112*x^6 + 5380*x^9 + 346788*x^12 + 27285968*x^15 + 2498963752*x^18 + 259124694312*x^21 + ... + a(n)*x^(3*n) + ...
%e A361543 where
%e A361543 A(x) = 1 + (d/dx x^4*A(x)^3) + (d^2/dx^2 x^8*A(x)^6)/2! + (d^3/dx^3 x^12*A(x)^9)/3! + (d^4/dx^4 x^16*A(x)^12)/4! + (d^5/dx^5 x^20*A(x)^15)/5! + ... + (d^n/dx^n x^(4*n)*A(x)^(3*n))/n! + ...
%e A361543 Related series.
%e A361543 Let B(x) = Series_Reversion(x - x^4*A(x)^3), which begins
%e A361543 B(x) = x + x^4 + 16*x^7 + 538*x^10 + 26676*x^13 + 1705373*x^16 + 131524408*x^19 + ... + A361310(n+1)*x^(3*n+1) + ...
%e A361543 then A(x) = B'(x) and
%e A361543 B(x) = x * exp( x^3*A(x)^3 + (d/dx x^7*A(x)^6)/2! + (d^2/dx^2 x^11*A(x)^9)/3! + (d^3/dx^3 x^15*A(x)^12)/4! + (d^4/dx^4 x^19*A(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A(x)^(3*n))/n! + ... ).
%t A361543 nmax = 20; r = 4; 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 A361543 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A361543 {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(4*m)*A^(3*m)/m!)) +O(x^(3*n+1))); polcoeff(A, 3*n)}
%o A361543 for(n=0, 25, print1(a(n), ", "))
%o A361543 (PARI) /* Using series reversion (faster) */
%o A361543 {a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^4*A^3 +O(x^(3*n+2))))); polcoeff(A, 3*n)}
%o A361543 for(n=0, 25, print1(a(n), ", "))
%Y A361543 Cf. A360950, A360974, A360975, A360973, A361046, A361536, A361537, A361541, A361542, A361551.
%Y A361543 Cf. A361310.
%K A361543 nonn
%O A361543 0,2
%A A361543 _Paul D. Hanna_, Mar 15 2023