OFFSET
0,3
COMMENTS
First negative term is a(77).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.
(1) A(x) = Sum_{n>=0} x^n * ( (exp(sqrt(n)*x) + x)^sqrt(n) + exp(n*x)/(1 + x*exp(sqrt(n)*x))^sqrt(n) )/2.
(2) A(x) = Sum_{n>=0} x^n * ( (exp(sqrt(n)*x) + x)^sqrt(n) + 1/(exp(-sqrt(n)*x) + x)^sqrt(n) )/2.
EXAMPLE
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 124*x^4/4! + 1015*x^5/5! + 10446*x^6/6! + 124894*x^7/7! + 1734160*x^8/8! + 27065133*x^9/9! + 473544010*x^10/10! + 9079863496*x^11/11! + 190885380192*x^12/12! + ...
PROG
(PARI) /* must set precision suitable for desired number of terms */
\p200
{a(n) = my(A=1); A = sum(m=0, n, x^m/2 * ( (exp(sqrt(m)*x +x*O(x^n)) + x)^sqrt(m) + exp(m*x +x*O(x^n))/(1 + x*exp(sqrt(m)*x +x*O(x^n)) )^sqrt(m) ) ); round(n!*polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jan 02 2023
STATUS
approved