login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A195514 E.g.f. satisfies: A(x) = exp(x/A(x)) - exp(x) + exp(x*A(x)). 3
1, 1, 1, 7, 49, 431, 5281, 69735, 1163681, 21106783, 443495041, 10238783303, 261253326577, 7289538431631, 220029116061665, 7180115829246391, 250782957484550977, 9365723147426913215, 371694221511705312769, 15638233703457023744679 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..280

FORMULA

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

EXAMPLE

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! +...

The e.g.f. of A195512 begins:

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).

The e.g.f. of A195513 begins:

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).

PROG

(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)}

CROSSREFS

Cf. A195512, A195513.

Sequence in context: A090016 A005924 A145358 * A204211 A125796 A135745

Adjacent sequences:  A195511 A195512 A195513 * A195515 A195516 A195517

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 20 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 5 21:00 EDT 2020. Contains 335473 sequences. (Running on oeis4.)