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A195514
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E.g.f. satisfies: A(x) = exp(x/A(x)) - exp(x) + exp(x*A(x)).
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3
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1, 1, 1, 7, 49, 431, 5281, 69735, 1163681, 21106783, 443495041, 10238783303, 261253326577, 7289538431631, 220029116061665, 7180115829246391, 250782957484550977, 9365723147426913215, 371694221511705312769, 15638233703457023744679
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OFFSET
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0,4
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LINKS
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FORMULA
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E.g.f. satisfies:
_ A(x) = Sum_{n>=0} x^n*(1/A(x)^n - 1 + A(x)^n) / n!.
_ A(x) = B(x/A(x)) where B(x) = A(x*B(x)) equals the e.g.f. of A195512 and satisfies: B(x) = exp(x) - exp(x*B(x)) + exp(x*B(x)^2).
_ A(x) = C(x/A(x)^2) where C(x) = A(x*C(x)^2) equals the e.g.f. of A195513 and satisfies: C(x) = exp(x*C(x)) - exp(x*C(x)^2) + exp(x*C(x)^3).
a(n) ~ sqrt((s^3*(exp(r/s) - exp(r)*s + exp(r*s)*s^2)) / (exp(r*s)*r*s^4 + exp(r/s)*(r + 2*s))) * n^(n-1) / (exp(n) * r^n), where r = 0.4188500786493484757126061423... and s = 2.29528345389537775167313... are roots of system of the equations exp(r) + s = exp(r/s) + exp(r*s), exp(r*s)*r = 1 + (exp(r/s)*r)/s^2. - Vaclav Kotesovec, Feb 26 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + x^2/2! + 7*x^3/3! + 49*x^4/4! + 431*x^5/5! +...
Related series begin:
exp(x/A(x)) = 1 + x - x^2/2! - 2*x^3/3! - 15*x^4/4! - 124*x^5/5! -...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 10*x^3/3! + 65*x^4/4! + 556*x^5/5! +...
B(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4426*x^5/5! +...
where A(x*B(x)) = B(x) = exp(x) - exp(x*B(x)) + exp(x*B(x)^2).
C(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! + 23471*x^5/5! +...
where A(x*C(x)^2) = C(x) = exp(x*C(x)) - exp(x*C(x)^2) + exp(x*C(x)^3).
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PROG
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(PARI) {a(n)=local(X=x+x*O(x^n), A=1+X); for(i=1, n, A=exp(X/A)-exp(X)+exp(X*A)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(X=x+x*O(x^n), A=1+X); for(i=1, n, A=sum(m=0, n, x^m*(1/A^m-1+A^m)/m!)); n!*polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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