OFFSET
0,4
COMMENTS
The sequence is surprisingly regular. Conjectures:
(1) a(4*n) and a(4*n+1) are odd, a(4*n+2) and a(4*n+3) are even, for n >= 1.
(2) a(n) is not divisible by 3 for n >= 0.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..125
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 161*x^5/5! + 1486*x^6/6! + 18922*x^7/7! + 268045*x^8/8! + 4609873*x^9/9! + 88133356*x^10/10! + ...
such that
A(x)^2 = exp(x) + x*exp(x*A(x)) + x^2*exp(x*A(x)^2)/2! + x^3*exp(x*A(x)^3)/3! + x^4*exp(x*A(x)^4)/4! + x^5*exp(x*A(x)^5)/5! + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 76*x^4/4! + 592*x^5/5! + 5794*x^6/6! + 70730*x^7/7! + 1019456*x^8/8! + 17176400*x^9/9! + 329143714*x^10/10! + ...
log(A(x)) = x + 3*x^3/3! + 6*x^4/4! + 100*x^5/5! + 645*x^6/6! + 10731*x^7/7! + 132328*x^8/8! + 2526336*x^9/9! + 46051785*x^10/10! + ...
PROG
(PARI) {a(n) = my(A=1 +x*O(x^n)); for(i=1, n,
A = sqrt( sum(m=0, n, x^m/m! * exp(x*A^m +x*O(x^n)) ) ));
n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 19 2021
STATUS
approved