|
| |
|
|
A200317
|
|
E.g.f. satisfies: A(x) = 1+x - cos(A(x)).
|
|
3
|
|
|
|
1, 1, 3, 14, 90, 736, 7308, 85364, 1146660, 17411296, 294880608, 5510730224, 112638576960, 2499645858256, 59850581734128, 1537891759461344, 42211161452391840, 1232537502720495616, 38148561442665067968, 1247578217573259535424, 42985394533206479112000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
Radius of convergence of e.g.f. A(x) is r = Pi/2-1 = 0.570796... where A(r) = Pi/2.
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f. satisfies:
(1) A(x) = Series_Reversion(x-1 + cos(x)).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (1-cos(x))^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1-cos(x))^(2*n)/x / n! ).
a(n) = ((n-1)!*sum(k=1..n-1, C(n+k-1,n-1)* sum(j=1..k, C(k,j)* ((sum(l=0..j-1, (C(j,l)* ((-1)^(n-l+j-1)+1) *sum(r=1..j-l, (C(j-l,r)*(-1)^(-r+(n-l+j-1)/2-l)* sum(i=0..(r-1)/2, (r-2*i)^(n-l+j-1)*C(r,i)))/2^r))/(n-l+j-1)!)))))), n>1, a(1)=1. [From Vladimir Kruchinin, Feb 20 2012]
a(n) ~ GAMMA(1/3) * n^(n-5/6) / (6^(1/6) * sqrt(Pi) * exp(n) * (Pi/2-1)^(n-1/3)). - Vaclav Kotesovec, Jan 18 2014
|
|
|
EXAMPLE
|
E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 14*x^4/4! + 90*x^5/5! +...
where A(x-1 + cos(x)) = x and A(x) = 1+x - cos(A(x)).
The e.g.f. satisfies:
A(x) = x + (1-cos(x)) + d/dx (1-cos(x))^2/2! + d^2/dx^2 (1-cos(x))^3/3! + d^3/dx^3 (1-cos(x))^4/4! +...
as well as the logarithmic series:
log(A(x)/x) = (1-cos(x))/x + d/dx (1-cos(x))^2/x/2! - d^2/dx^2 (1-cos(x))^3/x/3! + d^3/dx^3 (1-cos(x))^4/x/4! +...
|
|
|
MATHEMATICA
|
Rest[CoefficientList[InverseSeries[Series[x-1 + Cos[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 18 2014 *)
|
|
|
PROG
|
(PARI) {a(n)=n!*polcoeff(serreverse(x-1+cos(x+x^2*O(x^n))), n)}
for(n=1, 21, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (1-cos(x+x*O(x^n)))^m)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, (1-cos(x+x*O(x^n)))^m/x)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
(Maxima) a(n):=if n=1 then 1 else ((n-1)!*sum(binomial(n+k-1, n-1)* sum(binomial(k, j)*((sum((binomial(j, l)*((-1)^(n-l+j-1)+1)*sum((binomial(j-l, r) *(-1)^(-r+(n-l+j-1)/2-l)*sum((r-2*i)^(n-l+j-1)*binomial(r, i), i, 0, (r-1)/2))/2^r, r, 1, j-l))/(n-l+j-1)!, l, 0, j-1))), j, 1, k), k, 1, n-1)); [From Vladimir Kruchinin, Feb 20 2012]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,nice
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
