OFFSET
1,2
COMMENTS
f(x) satisfies f(x) = x * cos(f(x)), and the coefficients can be determined from this.
Alternatively, a(n) can be written in terms of (2*n)! as a(n) = (2*n)!/A309206(n).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..100
W. M. Gosper, Material from Bill Gosper's Computers & Math talk, M.I.T., 1989, i+38+1 pages, annotated and scanned, included with the author's permission. (There are many blank pages because about half of the original pages were two-sided, half were one-sided.)
W. M. Gosper, Superposition of plots -4<z<4 of z, cos z, z cos z, cos( z cos z), z cos(z cos z), ..., Jul 29 2019
EXAMPLE
f(x) = x - (1/2)*x^3 + (13/24)*x^5 - (541/720)*x^7 + (9509/8064)*x^9 - (7231801/3628800)*x^11 + (1695106117/479001600)*x^13 - (567547087381/87178291200)*x^15 + ...
Coefficients are
1, -1/2, 13/24, -541/720, 9509/8064, -7231801/3628800, 1695106117/479001600, -567547087381/87178291200, 36760132319047/2988969984000, -151856004814953841/ 6402373705728000, 113144789723082206461/2432902008176640000, ...
MAPLE
M := 20;
f := add(c[i]*z^(2*i+1), i=0..M);
f := series(f, z, 2*M+3);
f2 := series(z*cos(f)-f, z, 2*M+3);
for i from 0 to M do e[i]:=coeff(f2, z, 2*i+1); od:
elis:=[seq(e[i], i=0..M)]; clis:=[seq(c[i], i=0..M)];
t1 := solve(elis, clis); t2 := op(t1);
t3 := subs(t2, clis);
map(denom, t3);
# Alternative:
G:= f -> f - x*cos(f):
Newt:= unapply( f - G(f)/D(G)(f), f):
ff:= x:
for k from 1 to 5 do
ff:= convert(series(Newt(ff), x, 2^(k+1)), polynom)
od:
seq(denom(coeff(ff, x, 2*i+1), i=0..31); # Robert Israel, Aug 18 2019
MATHEMATICA
seq[n_] := Module[{p, k}, p = 1+O[x]; For[k = 2, k <= n, k++, p = Cos[x*p]]; p] // CoefficientList[#, x^2]& // Denominator;
seq[16] (* Jean-François Alcover, Sep 07 2019, from PARI *)
PROG
(PARI) \\ here F(n) gives n terms of power series.
F(n)={my(p=1+O(x)); for(k=2, n, p=cos(x*p)); p}
seq(n)={my(v=Vec(F(n))); vector(n, k, denominator(v[2*k-1]))} \\ Andrew Howroyd, Aug 17 2019
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Jul 28 2019
STATUS
approved