OFFSET
0,4
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..405
FORMULA
E.g.f.: A(x) = exp(x*exp(-x*exp(x*exp(-x*exp(x*...))))).
a(n+1) = Sum_{i=0..n} (i+1)*(-1)^i*binomial(n,i)*a(i)*a(n-i) - from a formula given in A096538 by Vladeta Jovovic.
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (n-k+1)^(k-1) * k^(n-k). - Paul D. Hanna, Jun 13 2009
|a(n)| ~ c * n! / (n^(3/2) * r^n), where r = 0.5098636055230131449434409623392631606695606770070519241... is the root of the equation r*exp(1/LambertW(-I/r))/I = LambertW(-I/r), and c = 0.63217617290426743984700577681768332... if n is even, and c = 1.4315233793609300008688492299361204... if n is odd. - Vaclav Kotesovec, Feb 26 2014
EXAMPLE
E.g.f.: A(x) = 1 + x - x^2/2! - 8*x^3/3! + 21*x^4/4! + 336*x^5/5! --++ ...
Log(A(x)) = x - x^2 - x^3/2! + 8*x^4/3! + 21*x^4/4! - 336*x^5/5! -++- ...
MATHEMATICA
Flatten[{1, Table[Sum[(-1)^(n-k) * Binomial[n, k] * (n-k+1)^(k-1) * k^(n-k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 26 2014 *)
PROG
(PARI) {a(n)=local(A=1); for(i=0, n, A=exp((-1)^(n-i)*x*A+x*O(x^n))); n!*polcoeff(A, n)}
(PARI) {a(n)=sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(n-k+1)^(k-1)*k^(n-k))} \\ Paul D. Hanna, Jun 13 2009
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 28 2008
STATUS
approved