OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^(3*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^4*A(x)^3).
(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.
(4) a(n) = (3*n+1) * A361310(n+1) for n >= 0.
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
EXAMPLE
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) + ...
where
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! + ...
Related series.
Let B(x) = Series_Reversion(x - x^4*A(x)^3), which begins
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) + ...
then A(x) = B'(x) and
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! + ... ).
MATHEMATICA
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 *)
PROG
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{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)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Using series reversion (faster) */
{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)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 15 2023
STATUS
approved