OFFSET
1,3
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)^2 ).
(2) A''(x) = A'(x)^4 / (1 - 2*A(x)*A'(x)^2).
(3) A'(x)^2 = Sum_{n>=0} 2^n * (n+1)^(n-1) * A(x)^n/n!.
(4) A(x)*A'(x)^2 = Sum_{n>=1} (2*n)^(n-1) * A(x)^n/n!.
(5) A(x) = x + Sum_{n>=2} (2*n-3)^(n-2) * A(x)^n/n!.
(6) A(x) = Series_Reversion( Integral sqrt( -2*x/LambertW(-2*x) ) dx ).
Let W(x) = Sum_{n>=2} (2*n-3)^(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.: Series_Reversion( (1 - exp(3*LambertW(-2*x)/2) * (1 + 3*LambertW(-2*x)))/9 ). - Shenghui Yang, Jan 24 2026
a(n) ~ 3^(2*n-3) * exp(n/2 - 1) * n^(n-2) / (sqrt(2) * (2 + exp(3/2))^(n - 3/2)). - Vaclav Kotesovec, Jan 24 2026
From Seiichi Manyama, Jun 16 2026: (Start)
For k >= 0, e.g.f. A(x) satisfies A'(x) = exp( A(x)*A'(x)^k ), with A(0)=0.
Let R(x) = Series_Reversion( (1-(1-k*(k+1)*x) * exp(-(k+1)*x))/(k+1)^2 ).
A(x) = R(x) * exp(-k*R(x)).
For k > 0, with V(x) = LambertW(e^(1/k) * (1-(k+1)^2*x)/k), we have R(x) = (1-k*V(x))/(k*(k+1)) and A(x) = (1-k*V(x))/(k*(k+1)) * exp((k*V(x)-1)/(k+1)). (End)
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 6*x^3/3! + 70*x^4/4! + 1228*x^5/5! + 28884*x^6/6! + 853792*x^7/7! + 30426112*x^8/8! + 1270005984*x^9/9! + ...
By definition log(A'(x)) = A(x)*A'(x)^2, where:
(3) A'(x)^2 = 1 + 2*x + 14*x^2/2! + 176*x^3/3! + 3232*x^4/4! + 78448*x^5/5! + 2373232*x^6/6! + 86101280*x^7/7! + 3645713312*x^8/8! + ...
(4) A(x)*A'(x)^2 = x + 5*x^2/2! + 54*x^3/3! + 906*x^4/4! + 20688*x^5/5! + 598608*x^6/6! + 20989152*x^7/7! + 864968544*x^8/8! + 40974175296*x^9/9! + ...
RELATED SERIES:
Let W(x) = x^2/2! + 3*x^3/3! + 25*x^4/4! + 343*x^5/5! + 6561*x^6/6! + 161051*x^7/7! + ... + (2*n-3)^(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[-(1/9) (E^ProductLog[-2 x])^(3/2) (1+3 ProductLog[-2 x]), {x, 0, max}]]//Normal, (x)->u-1/9, Infinity], u]]*Range[max]! (* Shenghui Yang, Jan 24 2026 *)
PROG
(PARI) \\ By definition A'(x) = exp( A(x)*A'(x)^2 )
{a(n) = my(A=x); for(i=1, n, A = intformal(exp(A*A'^2 +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, (2*m-3)^(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, (2*m-3)^(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), ", "))
(PARI) my(k=2, N=20, x='x+O('x^N), R=serreverse((1-(1-k*(k+1)*x)*exp(-(k+1)*x))/(k+1)^2)); Vec(serlaplace(R*exp(-k*R))) \\ Seiichi Manyama, Jun 16 2026
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Jan 23 2026
STATUS
approved