OFFSET
0,3
FORMULA
E.g.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( exp(n*x) - A(x) )^n.
(2) 1 = Sum_{n>=0} exp(n^2*x) / (1 + exp(n*x)*A(x))^(n+1).
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 55*x^3/3! + 2439*x^4/4! + 181711*x^5/5! + 19987863*x^6/6! + 3019344175*x^7/7! + 597283032279*x^8/8! + 149571915236911*x^9/9! + 46218017081300823*x^10/10! + ...
such that
1 = 1 + (exp(x) - A(x)) + (exp(2*x) - A(x))^2 + (exp(3*x) - A(x))^3 + (exp(4*x) - A(x))^4 + (exp(5*x) - A(x))^5 + (exp(6*x) - A(x))^6 + (exp(7*x) - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + exp(x)/(1 + exp(x)*A(x))^2 + exp(4*x)/(1 + exp(2*x)*A(x))^3 + exp(9*x)/(1 + exp(3*x)*A(x))^4 + exp(16*x)/(1 + exp(4*x)*A(x))^5 + exp(25*x)/(1 + exp(5*x)*A(x))^6 + exp(36*x)/(1 + exp(6*x)*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = x + 2*x^2/2! + 48*x^3/3! + 2222*x^4/4! + 169080*x^5/5! + 18843302*x^6/6! + 2872307088*x^7/7! + 571992255662*x^8/8! + 143972732107560*x^9/9! + 44668284142577462*x^10/10! + ...
The derivative of e.g.f. A(x) equals the ratio of the series:
A'(x) = [ Sum_{n>=1} n^2 * exp(n*x) * ( exp(n*x) - A(x) )^(n-1) ] / [ Sum_{n>=1} n * ( exp(n*x) - A(x) )^(n-1) ]. - Paul D. Hanna, Aug 06 2018
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (exp(m*x +x*O(x^#A)) - Ser(A))^m ) )[#A] ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 16 2018
STATUS
approved