A141369 E.g.f. satisfies A(x) = exp(x*A(-x)).

1, 1, -1, -8, 21, 336, -1445, -35328, 212009, 7010560, -54073449, -2258780160, 21303275389, 1076400869376, -12005345614093, -712084337721344, 9169911825026385, 624667885401341952, -9122376282532978769, -701910552416102645760, 11462725659070874233061

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

Cf. A096538.

Sequence in context: A228504 A270552 A156239 * A060390 A019281 A284737

Adjacent sequences: A141366 A141367 A141368 * A141370 A141371 A141372

Keyword

sign

Author

Paul D. Hanna, Jun 28 2008

Status

approved