OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..160
FORMULA
E.g.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - exp(-n*x) )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - exp(-(n+1)*x) )^n.
(3) 1 = Sum_{n>=0} exp(-(n+1)*x) * ( 1/A(x) - exp(-(n+1)*x) )^n.
(4) A(x)^2 = 2*A(x) * [ Sum_{n>=0} (n+1) * ( 1/A(x) - exp(-(n+1)*x) )^n ] - [ Sum_{n>=0} (n+1) * ( 1/A(x) - exp(-(n+2)*x) )^n ].
(5) A(x) = [ Sum_{n>=1} n*(n+1)/2 * exp(-(n+1)*x) * ( 1/Ser(A) - exp(-(n+1)*x) )^(n-1) ] / [ Sum_{n>=1} n^2 * exp(-n*x) * ( 1/Ser(A) - exp(-n*x) )^(n-1) ].
a(n) ~ sqrt(Pi) * n^(2*n + 1/2) / (4*sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 10 2018
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 783*x^4/4! + 35551*x^5/5! + 2465943*x^6/6! + 238958791*x^7/7! + 30604867023*x^8/8! + 4988281843471*x^9/9! + ...
such that
1 = 1 + (1/A(x) - exp(-x)) + (1/A(x) - exp(-2*x))^2 + (1/A(x) - exp(-3*x))^3 + (1/A(x) - exp(-4*x))^4 + (1/A(x) - exp(-5*x))^5 + (1/A(x) - exp(-6*x))^6 + (1/A(x) - exp(-7*x))^7 + (1/A(x) - exp(-8*x))^8 + ...
Also,
A(x) = 1 + (1/A(x) - exp(-2*x)) + (1/A(x) - exp(-3*x))^2 + (1/A(x) - exp(-4*x))^3 + (1/A(x) - exp(-5*x))^4 + (1/A(x) - exp(-6*x))^5 + (1/A(x) - exp(-7*x))^6 + (1/A(x) - exp(-8*x))^7 + (1/A(x) - exp(-9*x))^8 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - exp(-(m+1)*x +x*O(x^#A)) )^m ) )[#A]/2 ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 02 2018
STATUS
approved