OFFSET
1,3
LINKS
Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 1..200 (first 150 terms from Paul D. Hanna)
FORMULA
a(n) = A274275(n)/n.
E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / (n-1)! satisfies:
(1) A( sqrt( A(x^2*exp(2*x)) ) ) = -LambertW(-x*exp(x)).
(2) A(x) = Series_Reversion( sqrt( A(x^2*exp(-x)) ) ).
(3) A( A(x)^2 * exp(-2*A(x)) ) = x^2.
(4) A(-A(x)^2 * exp(-2*A(x)) ) = -LambertW(x^2*exp(-x^2)).
a(n)/n! ~ c * d^n / n^(5/2), where d = 2.52462188117..., c = 0.36965356... . - Vaclav Kotesovec, Jun 23 2016
EXAMPLE
E.g.f.: A(x) = x + x^2 + 2*x^3/2! + 10*x^4/3! + 80*x^5/4! + 776*x^6/5! + 8992*x^7/6! + 130768*x^8/7! + 2252672*x^9/8! + 43823872*x^10/9! + 957193856*x^11/10! +...
such that A( sqrt( A(x^2*exp(-2*x)) ) ) = x.
PROG
(PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse( sqrt( subst(A, x, x^2*exp(-2*x +x*O(x^n))) ) ) ); (n-1)!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 17 2016
STATUS
approved