OFFSET
0,2
COMMENTS
Conjecture: If r>=2 and s>=1 and A(x) = Sum_{n>=0} d^n/dx^n x^(r*n) * A(x)^(s*n) / n!, then a(n) ~ c(r,s) * n! * n^alpha(r,s) * ((r-1)/LambertW(1/s))^n, where alpha(r,s) = ((2*s+1)*LambertW(1/s) + 1 + 1/(1 + LambertW(1/s))) * r/(2*(r-1)) - 1 and c(r,s) is a constant independent on n. - Vaclav Kotesovec, Mar 16 2023
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^(4*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(5*n) * A(x)^n / n!.
(2) A(x) = d/dx Series_Reversion(x - x^5*A(x)).
(3) B(x - x^5*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(5*n-1) * A(x)^n / n! ) is the g.f. of A361311.
(4) a(n) = (4*n+1) * A361311(n+1) for n >= 0.
a(n) ~ c * 4^n * n! * n^((15*LambertW(1) - 3 + 5/(1 + LambertW(1)))/8) / LambertW(1)^n, where c = 0.438413009716541247480505206... - Vaclav Kotesovec, Mar 16 2023
EXAMPLE
G.f.: A(x) = 1 + 5*x^4 + 90*x^8 + 2535*x^12 + 93840*x^16 + 4226355*x^20 + 222038775*x^24 + 13259599965*x^28 + ... + a(n)*x^(4*n) + ...
where
A(x) = 1 + (d/dx x^5*A(x)) + (d^2/dx^2 x^10*A(x)^2)/2! + (d^3/dx^3 x^15*A(x)^3)/3! + (d^4/dx^4 x^20*A(x)^4)/4! + (d^5/dx^5 x^25*A(x)^5)/5! + ... + (d^n/dx^n x^(5*n)*A(x)^n)/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^5*A(x)), which begins
B(x) = x + x^5 + 10*x^9 + 195*x^13 + 5520*x^17 + 201255*x^21 + 8881551*x^25 + ... + A361311(n+1)*x^(4*n+1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^4*A(x) + (d/dx x^9*A(x)^2)/2! + (d^2/dx^2 x^14*A(x)^3)/3! + (d^3/dx^3 x^19*A(x)^4)/4! + (d^4/dx^4 x^24*A(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(5*n-1)*A(x)^n)/n! + ... ).
MATHEMATICA
nmax = 20; r = 5; 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 *)
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^(5*m)*A^(1*m)/m!)) +O(x^(4*n+1))); polcoeff(A, 4*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^5*A^1 +O(x^(4*n+2))))); polcoeff(A, 4*n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 15 2023
STATUS
approved