A317348 E.g.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - exp(-n*x) )^n = 1.

1, 1, 3, 31, 783, 35551, 2465943, 238958791, 30604867023, 4988281843471, 1006426188747783, 246050857141536151, 71658459729884788863, 24512979124556543501791, 9733113984959380709677623, 4440214540533789234079579111, 2306721251730615059447461056303, 1354037785009235729190621178158511

Offset: 0

Author

Paul D. Hanna, Aug 02 2018

Keywords

• nonn

Examples

			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  + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..160

Crossrefs

Cf. A317349.

Programs

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), ", "))

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