OFFSET
1,3
COMMENTS
Conjecture: a(n) is even iff n is divisible by 3 for n >= 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
(1) A'(x) = exp( A(x)*A'(x)^3 ).
(2) A''(x) = A'(x)^5 / (1 - 3*A(x)*A'(x)^3).
(3) A'(x)^3 = Sum_{n>=0} 3^n * (n+1)^(n-1) * A(x)^n/n!.
(4) A(x)*A'(x)^3 = Sum_{n>=1} (3*n)^(n-1) * A(x)^n/n!.
(5) A(x) = x + Sum_{n>=2} (3*n-4)^(n-2) * A(x)^n/n!.
(6) A(x) = Series_Reversion( Integral ( -3*x/LambertW(-3*x) )^(1/3) dx ).
Let W(x) = Sum_{n>=2} (3*n-4)^(n-2) * x^n/n!, then e.g.f. A(x) satisfies:
(7) A(x) = Series_Reversion(x - W(x)).
(8) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) W(x)^n/n!.
(9) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) W(x)^n/(n!*x) ).
E.g.f. A(x) = Series_Reversion( (1 - exp(4*LambertW(-3*x)/3) * (1 + 4*LambertW(-3*x)))/16 ). - Shenghui Yang, Jan 24 2026
a(n) ~ 2^(4*n-6) * exp(n/3 - 1) * n^(n-2) / (sqrt(3) * (3 + exp(4/3))^(n - 3/2)). - Vaclav Kotesovec, Jan 24 2026
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 8*x^3/3! + 129*x^4/4! + 3171*x^5/5! + 105252*x^6/6! + 4408983*x^7/7! + 223281351*x^8/8! + 13270085748*x^9/9! + ...
By definition log(A'(x)) = A(x)*A'(x)^3, where:
(3) A'(x)^3 = 1 + 3*x + 30*x^2/2! + 537*x^3/3! + 14049*x^4/4! + 486306*x^5/5! + 21000591*x^6/6! + 1088451837*x^7/7! + 65883123414*x^8/8! + ...
(4) A(x)*A'(x)^3 = x + 7*x^2/2! + 107*x^3/3! + 2553*x^4/4! + 83121*x^5/5! + 3434871*x^6/6! + 172191843*x^7/7! + 10153225737*x^8/8! + 688570504569*x^9/9! + ...
RELATED SERIES:
Let W(x) = x^2/2! + 5*x^3/3! + 64*x^4/4! + 1331*x^5/5! + 38416*x^6/6! + 1419857*x^7/7! + ... + (3*n-4)^(n-2)*x^n/n! + ... then
(7) A(x) = x + W(A(x)); equivalently, A(x - W(x)) = x.
(8) A(x) = x + W(x) + d/dx W(x)^2/2! + d^2/dx^2 W(x)^3/3! + d^3/dx^3 W(x)^4/4! + ...
(9) log(A(x)/x) = W(x)/x + d/dx W(x)^2/(2!*x) + d^2/dx^2 W(x)^3/(3!*x) + d^3/dx^3 W(x)^4/(4!*x) + ...
MATHEMATICA
max=30; Rest[CoefficientList[Replace[InverseSeries[Series[-(1/16) (E^ProductLog[-3 x])^(4/3) (1+4 ProductLog[-3 x]), {x, 0, max}]]//Normal, (x)->u-1/16, Infinity], u]]*Range[max]! (* Shenghui Yang, Jan 24 2026 *)
PROG
(PARI) \\ By definition A'(x) = exp( A(x)*A'(x)^3 )
{a(n) = my(A=x); for(i=1, n, A = intformal(exp(A*A'^3 +x*O(x^n)))); n!*polcoef(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) \\ Formula (7)
{a(n) = my(A=x, W = sum(m=2, n+1, (3*m-4)^(m-2)*x^m/m!) +x*O(x^n));
A = serreverse(x - W); n!*polcoef(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) \\ Formula (8)
{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x, W = sum(m=2, n+1, (3*m-4)^(m-2)*x^m/m!) +x*O(x^n));
A = x + sum(m=1, n, Dx(m-1, W^m/m!)); n!*polcoef(A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 24 2026
STATUS
approved