OFFSET
1,2
FORMULA
a(n) = 2*A143138(n) for n>1.
E.g.f. A(x) satisfies:
(1) A(x) = x + 2 * (exp( (A(x) + x)/2 ) - 1)^2.
(2) A(x) = -x + 2 * Series_Reversion( x - (exp(x)-1)^2 ).
(3) A(x) = x + 2 * Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^(2*n) / n!.
(4) A( log(1+x) - x^2 ) = log(1+x) + x^2.
EXAMPLE
E.g.f.: A(x) = x + 4*x^2/2! + 36*x^3/3! + 508*x^4/4! + 10020*x^5/5! + 253804*x^6/6! + 7853076*x^7/7! + 287078908*x^8/8! + 12106864260*x^9/9! + 578586544204*x^10/10! +...
such that A( x - (exp(x) - 1)^2 ) = x + (exp(x) - 1)^2.
PROG
(PARI) {a(n) = n!*polcoeff( -x + 2*serreverse( x - (exp(x +x*O(x^n)) - 1)^2 ), n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 15 2016
STATUS
approved