|
| |
|
|
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
|
|
|
|
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
