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A317356
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E.g.f. satisfies: A(x) = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n / (2 - exp(n*x)*A(x))^(n+1).
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4
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1, 2, 14, 134, 4358, 589622, 102434534, 21285122294, 5530748479718, 1792785367579382, 711595226383338854, 339665400624638782454, 192071493764203628322278, 127053485326157331378577142, 97253813187878484942034153574, 85330814329687863076988482842614, 85104598195236153766017309663096038
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OFFSET
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0,2
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COMMENTS
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E.g.f. A(x) = G(exp(x) - 1), where G(x) is the g.f. of A317351.
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 0..136
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FORMULA
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E.g.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n / (2 - exp(n*x)*A(x))^(n+1),
(2) A(x) = Sum_{n>=0} ( exp((n+1)*x) + A(x) )^n / (2 + exp(n*x)*A(x))^(n+1).
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.2625457134... - Vaclav Kotesovec, Aug 10 2018
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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 134*x^3/3! + 4358*x^4/4! + 589622*x^5/5! + 102434534*x^6/6! + 21285122294*x^7/7! + 5530748479718*x^8/8! + 1792785367579382*x^9/9! + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A) + (exp(2*x) - A)/(2 - exp(x)*A)^2 + (exp(3*x) - A)^2/(2 - exp(2*x)*A)^3 + (exp(4*x) - A)^3/(2 - exp(3*x)*A)^4 + (exp(5*x) - A)^4/(2 - exp(4*x)*A)^5 + (exp(6*x) - A)^5/(2 - exp(5*x)*A)^6 + ...
Also,
A(x) = 1/(2 + A) + (exp(2*x) + A)/(2 + exp(x)*A)^2 + (exp(3*x) + A)^2/(2 + exp(2*x)*A)^3 + (exp(4*x) + A)^3/(2 + exp(3*x)*A)^4 + (exp(5*x) + A)^4/(2 + exp(4*x)*A)^5 + (exp(6*x) + A)^5/(2 + exp(5*x)*A)^6 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( exp((m+1)*x +x*O(x^#A)) - Ser(A) )^m / (2 - exp(m*x +x*O(x^#A))*Ser(A))^(m+1) ) ) ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))
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CROSSREFS
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Cf. A317356, A317351.
Sequence in context: A306081 A326886 A111424 * A336182 A224729 A355722
Adjacent sequences: A317353 A317354 A317355 * A317357 A317358 A317359
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Aug 02 2018
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STATUS
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approved
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