OFFSET
0,3
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 96*x^4/4! + 1120*x^5/5! + 16260*x^6/6! + 290640*x^7/7! + 6108480*x^8/8! + 148353408*x^9/9! + ...
where
exp(x*A(x) + x/A(x)) = 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! +...
explicitly,
exp(x*A(x) + x/A(x)) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 88*x^4/4! + 872*x^5/5! + 11464*x^6/6! + 189968*x^7/7! + 3774208*x^8/8! + 87674336*x^9/9! + ...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 97*x^4/4! + 981*x^5/5! + 13291*x^6/6! + 222013*x^7/7! + 4458273*x^8/8! + 104169817*x^9/9! + ...
PROG
(PARI) {a(n) = my(A=1+x +x^3*O(x^n), H=A);
for(k=1, n, A = A*exp(-x*A)*exp(-x/A) * sum(m=0, n+3, x^m/m! * exp(x*A^m +x^3*O(x^n)) );
A = truncate( H + polcoeff(A, k+2)*x^k ) +x^3*O(x^n); H=A); n!*polcoeff(W=A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 08 2021
STATUS
approved