A143600 E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)).

1, 1, 5, 25, 249, 2561, 40573, 641817, 14110001, 302279617, 8530496181, 230851019609, 7964867290537, 260618470319169, 10635790073585069, 408342804482252761, 19246730825243728737, 848289638051491455617, 45356940470607637151845, 2257054105205570995111833

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..175

Formula

E.g.f. A(x) satisfies:

(1) A(x) = exp(x*exp(2x*G(2x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.

(2) [A(x)/A(-x) + A(-x)/A(x)]/2 = G(2x) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.

(3) A(x)/A(-x) = exp(x*[A(x)/A(-x) + A(-x)/A(x)]) = F(2x) where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.

(4) A(x) = Sum_{n>=0} (n+1)^(n-1) * x^n/n! / A(-x)^n.

(5) A(x)^m = Sum_{n>=0} m*(n+m)^(n-1) * x^n/n! / A(-x)^n.

(6) log(A(x)) = Sum_{n>=1} n^(n-1) * x^n/n! / A(-x)^n = x*A(x)/A(-x).

Formulas (4), (5), and (6) are due to LambertW identities. - Paul D. Hanna, Nov 05 2012

a(n) ~ c * n! / (n^(3/2) * r^n), where r = 0.33137170967459079... is the root of the equation sqrt(1+4*r^2) = log((1+sqrt(1+4*r^2))/(2*r)), and c = 1.35397895306096963692514418... if n is even, and c = 1.281887793570420328585518150... if n is odd. - Vaclav Kotesovec, Feb 25 2014

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +...
A LambertW identity yields the series:
A(x) = 1 + x/A(-x) + 3^1*x^2/2!/A(-x)^2 + 4^2*x^3/3!/A(-x)^3 + 5^3*x^4/4!/A(-x)^4 + 6^4*x^5/5!/A(-x)^5 +...+ (n+1)^(n-1)*x^n/n!/A(-x)^n +...
RELATED EXPANSIONS.
A(x)/A(-x) = F(2x) where F(x) is the e.g.f. of A058014:
A(x)/A(-x) = 1 + 2*x + 4*x^2/2! + 32*x^3/3! + 208*x^4/4! + 3072*x^5/5! +...
F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
which satisfies: F(x) = exp(x*(F(x) + 1/F(x))/2).
(A(x)/A(-x) + A(-x)/A(x))/2 = G(2x) where G(x) is the e.g.f. of A143601:
(A(x)/A(-x) + A(-x)/A(x))/2 = 1 + 4*x^2/2! + 208*x^4/4! + 34624*x^6/6! +...
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
which satisfies G(x) = cosh(x*G(x)).

Prog

(PARI) a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=exp(x*A/subst(A, x, -x))); n!*polcoeff(A, n)
(PARI) /* Formula Using a LambertW Identity: */
{a(n)=local(A=1); for(i=1, n, A=sum(k=0, n, (k+1)^(k-1)*x^k/k!/subst(A, x, -x)^k+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 05 2012

Crossrefs

Cf. A058014, A143601, A007106.

Sequence in context: A327619 A005452 A061839 * A209529 A184958 A145076

Adjacent sequences: A143597 A143598 A143599 * A143601 A143602 A143603

Keyword

nonn

Author

Paul D. Hanna, Aug 26 2008

Status

approved