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A143599 E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ). 2
1, 1, 3, 10, 53, 316, 2527, 22072, 239689, 2774800, 38284091, 553477024, 9284250109, 161180444608, 3187413648343, 64638167906176, 1473221217774353, 34190645940363520, 882759869810501491, 23079229227696318976 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..218

FORMULA

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

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

E.g.f.: [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601.

E.g.f.: [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106.

E.g.f.: A(x) = H(x/2)^2 where H(x) = exp(x*H(x)/H(-x)) = e.g.f. of A143600.

E.g.f. satisfies: A(x/cosh(x)) = exp(x*exp(x)/cosh(x)). [From Paul D. Hanna, Aug 29 2008]

E.g.f. satisfies: -x*sqrt(x^2/log(y)^2) = log((x^2*y)/log(y)^2), where y=A(x). - Vaclav Kotesovec, Feb 26 2014

a(n) ~ c * n! * d^n / n^(3/2), where d = 1.5088795615383199289... is the root of the equation sqrt(1+1/d^2) = 1 + LambertW((1+sqrt(1+1/d^2))/exp(1 + sqrt(1+1/d^2))), and c = 7.98255033020099890281693169... if n is even, and c = 7.852067808737280621088934789... if n is odd. - Vaclav Kotesovec, Feb 26 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...

F(x) = sqrt(A(x)/A(-x)) = e.g.f. of A058014:

F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...

where F(x) = exp(x*(F(x) + 1/F(x))/2).

G(x) = [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601:

G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...

where G(x) = cosh(x*G(x)).

S(x) = [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106:

S(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...

where S(x) = sqrt(G(x)^2 - 1) and G(x) = e.g.f. of A143601.

PROG

(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=exp(x*sqrt(A/subst(A, x, -x)))); n!*polcoeff(A, n)}

CROSSREFS

Cf. A058014, A143600, A143601, A007106.

Sequence in context: A042171 A133148 A189815 * A264409 A199202 A135829

Adjacent sequences:  A143596 A143597 A143598 * A143600 A143601 A143602

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 27 2008

STATUS

approved

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Last modified July 18 11:23 EDT 2019. Contains 325138 sequences. (Running on oeis4.)