A309205 -id:A309205 - cp's OEIS frontend

A309206 a(n) = (2*n)!/A309205(n).

1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 13, 19, 5, 1, 221, 1, 1, 1, 1, 1, 13, 17, 5, 1, 47, 4913, 29, 7, 11, 53, 1, 47, 325, 13, 1147, 41, 1, 1, 41, 1081, 11, 1, 5, 1, 1, 83, 1, 1, 133, 1, 2491, 97, 5, 103, 61, 1, 1, 19, 226493, 1, 1, 1, 5, 31, 1, 1, 1, 1271, 289

Offset: 0

N. J. A. Sloane, Jul 28 2019, following a suggestion from Bill Gosper

Bill Gosper points out that this is a better fingerprint for the series than A309205.

Cf. A309204, A309205.

F[n_] := Module[{p}, p = 1 + O[x]; For[k=2, k <= n, k++, p = Cos[x p]]; p];
seq[n_] := Module[{v}, v = CoefficientList[F[n], x]; Table[(2(k - 1))!/ Denominator[v[[2k - 1]]], {k, 1, n}]];
seq[71] (* Jean-François Alcover, Aug 27 2019, from 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, (2*(k-1))!/denominator(v[2*k-1]))} \\ Andrew Howroyd, Aug 17 2019

Terms a(31) and beyond from Andrew Howroyd, Aug 17 2019

A309204 Numerators of coefficients of odd powers of x in expansion of f(x) = x cos (x cos (x cos( ... .

Original entry on oeis.org

1, -1, 13, -541, 9509, -7231801, 1695106117, -567547087381, 36760132319047, -151856004814953841, 113144789723082206461, -103890621918675777804301, 8866964955352146292017421, -8002609021370033485261033939, 47038068678960604511245887564401, -421635069078222570953208470234640901

Offset: 1

N. J. A. Sloane, Jul 28 2019

f(x) satisfies f(x) = x * cos(f(x)), and the coefficients can be determined from this.

			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, ...

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

Cf. A309205.

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(numer, t3);

seq[n_] := Module[{p, k}, p = 1 + O[x]; For[k = 2, k <= n, k++, p = Cos[x*p]]; p] // CoefficientList[#, x^2] & // Numerator;
seq[16] (* Jean-François Alcover, Sep 07 2019, from PARI *)
Table[1/(2*n + 1)* SeriesCoefficient[Cos[t]^(2*n + 1), {t, 0, 2*n}], {n, 0, 15}] // Numerator; (* Christophe Vignat, Jan 06 2020 *)
Table[1/(2*n + 1)*Residue[(Cos[z]/z)^(2*n + 1), {z, 0}], {n, 0, 15}] // Numerator; (* Christophe Vignat, Jan 06 2020 *)

\\ 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, numerator(v[2*k-1]))} \\ Andrew Howroyd, Aug 17 2019

From Christophe Vignat, Jan 06 2020: (Start)
Numerator of (-1)^n/(2^(2*n+1)*(2*n+1)!)*Sum_{k=0..2*n+1} binomial(2*n+1,k)*(2*k-2*n-1)^(2*n).
Numerator of 1/(2*n+1)*(coefficient of t^(2*n) in cos(t)^(2*n+1)).
Numerator of 1/(2*n+1)*(residue of (cos(t)/t)^(2*n+1) at t=0). (End)

cp's OEIS Frontend

A309206 a(n) = (2*n)!/A309205(n).

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A309204 Numerators of coefficients of odd powers of x in expansion of f(x) = x cos (x cos (x cos( ... .

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