|
| |
|
|
A212913
|
|
E.g.f. satisfies: A(x) = exp( Integral 1 + x*A(x)^2 dx ), where the constant of integration is zero.
|
|
3
|
|
|
|
1, 1, 2, 8, 44, 308, 2648, 26912, 315536, 4193744, 62302496, 1023057536, 18400342208, 359733922880, 7595810693504, 172270928222720, 4176595617747200, 107793463235860736, 2950683535353324032, 85386983313510877184, 2604521649171407301632, 83519383797513832420352
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Compare to the identities:
(1) F(x) = exp( Integral 1 + x*F(x) dx ) when F(x) = 1/(1-x),
(2) G(x) = exp( Integral x*G(x)^2 dx ) when G(x) = 1/(1-x^2)^(1/2).
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f.: sqrt(2)*exp(x)/sqrt(exp(2*x) - 2*exp(2*x)*x + 1). - Vaclav Kotesovec, Jan 05 2014
a(n) ~ 2^(n+1) * n^n / (exp(n) * (1+LambertW(exp(-1)))^(n+1)). - Vaclav Kotesovec, Jan 05 2014
|
|
|
EXAMPLE
|
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 44*x^4/4! + 308*x^5/5! + 2648*x^6/6! +...
such that, by definition,
log(A(x)) = x + x^2/2! + 4*x^3/3! + 18*x^4/4! + 112*x^5/5! + 880*x^6/6! + 8256*x^7/7! +...
Related expansions:
x*A(x)^2 = x + 4*x^2/2! + 18*x^3/3! + 112*x^4/4! + 880*x^5/5! + 8256*x^6/6! +...
A(x)^2 = 1 + 2*x + 6*x^2/2! + 28*x^3/3! + 176*x^4/4! + 1376*x^5/5! + 12912*x^6/6! +...
|
|
|
MATHEMATICA
|
CoefficientList[Series[Sqrt[2]*E^x/Sqrt[E^(2*x) - 2*E^(2*x)*x + 1], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 05 2014 *)
|
|
|
PROG
|
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(intformal(1+x*A^2)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
