OFFSET
1,2
FORMULA
E.g.f.: Series_Reversion( Sum_{n>=1} (-1)^(n-1) * x^n / (n + x^n) ).
EXAMPLE
E.g.f.: A(x) = x + 3*x^2/2! + 19*x^3/3! + 189*x^4/4! + 2521*x^5/5! + 41813*x^6/6! + 826099*x^7/7! + 18889209*x^8/8! + 489450841*x^9/9! + 14142037665*x^10/10! + 449838952795*x^11/11! + ...
such that
x = A(x)/(1 + A(x)) + A(x)^2/(2 + A(x)^2) + A(x)^3/(3 + A(x)^3) + A(x)^4/(4 + A(x)^4) + A(x)^5/(5 + A(x)^5) + A(x)^6/(6 + A(x)^6) + ...
The series reversion of the e.g.f. begins:
Series_Reversion(A(x)) = x - 3*x^2/2! + 8*x^3/3! - 24*x^4/4! + 144*x^5/5! - 1010*x^6/6! + 5760*x^7/7! - 40320*x^8/8! + 416640*x^9/9! - 4250232*x^10/10! + 43545600*x^11/11! - 511526400*x^12/12! + ...
which equals Sum_{n>=1} (-1)^(n-1) * x^n / (n + x^n).
PROG
(PARI) {a(n) = my(A = serreverse( sum(m=1, n+1, (-1)^(m-1)*x^m/(m + x^m +x*O(x^n))) ));
n!*polcoeff(H=A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 26 2019
STATUS
approved