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^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)^3).
(3) B(x - x^2*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^(3*n) / n! ) is the g.f. of A360978.
(4) a(n) = (n+1) * A360978(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1/3)^n, where alfa = 2.598541481443... and c = 0.058191982295165... - Vaclav Kotesovec, Feb 28 2023
alfa = 7*LambertW(1/3) + 1/(1 + LambertW(1/3)). - Vaclav Kotesovec, Mar 13 2023
EXAMPLE
G.f.: A(x) = 1 + 2*x + 24*x^2 + 476*x^3 + 12380*x^4 + 386220*x^5 + 13821276*x^6 + 552876504*x^7 + 24318017424*x^8 + ... + a(n)*x^n + ...
where
A(x) = 1 + (d/dx x^2*A(x)^3) + (d^2/dx^2 x^4*A(x)^6)/2! + (d^3/dx^3 x^6*A(x)^9)/3! + (d^4/dx^4 x^8*A(x)^12)/4! + (d^5/dx^5 x^10*A(x)^15)/5! + (d^6/dx^6 x^12*A(x)^18)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^(3*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)^3), which begins
B(x) = x + x^2 + 8*x^3 + 119*x^4 + 2476*x^5 + 64370*x^6 + 1974468*x^7 + 69109563*x^8 + 2702001936*x^9 + ... + A360978(n)*x^n + ...
then A(x) = B'(x) and
B(x) = x * exp( x*A(x)^3 + (d/dx x^3*A(x)^6)/2! + (d^2/dx^2 x^5*A(x)^9)/3! + (d^3/dx^3 x^7*A(x)^12)/4! + (d^4/dx^4 x^9*A(x)^15)/5! + (d^5/dx^5 x^11*A(x)^18)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^(3*n))/n! + ... ).
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^(2*m)*A^(3*m)/m!)) +O(x^(n+1))); polcoeff(A, 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^2*A^3 +O(x^(n+2))))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 27 2023
STATUS
approved