A196873 E.g.f. A(x) satisfies: A(x*cos(x)) = 1/cos(x).

1, 1, 17, 961, 116129, 24419521, 7935695921, 3689986634881, 2325980044877633, 1909820766285110401, 1980859561990210210385, 2532925102150057552027201, 3914882960190010585911806177, 7195136206495188167173611850561, 15509756436233434005672607162123889

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..200

Formula

a(n) = (2*n)! * [x^(2*n)] (1/cos(x)^(2*n+1)) / (2*n+1).

a(n) = A185142(n) / (2*n+1).

E.g.f. satisfies:

(1) A(x) = 1/cos(x*A(x)).

(2) A(x) = Sum_{n>=0} A000364(n)*(x*A(x))^(2*n)/(2*n)!, where A000364(n) = E_{2*n} forms the Euler numbers.

(3) A(x) = (1/x)*Series_Reversion(x*cos(x)).

(4) A(x) = Sum_{n>=1} (-1)^(n-1)*4*(2*n-1)*Pi / ((2*n-1)^2*Pi^2 - 4*x^2*A(x)^2).

a(n) ~ s*sqrt(2/(1+r^2*s^2*(3*s^2-1))) * 2^(2*n) * n^(2*n-1) / (exp(2*n) * r^(2*n)), where s = 1.533308151311528971... is the root of the equation s*cos(1/sqrt(s^2-1)) = 1, and r = 1/(s*sqrt(s^2-1)) = 0.561096338191045067... - Vaclav Kotesovec, Jan 23 2014

Example

E.g.f.: A(x) = 1 + x^2/2! + 17*x^4/4! + 961*x^6/6! + 116129*x^8/8! + ...
where A(x) = 4*Pi/(Pi^2 - 4*x^2*A(x)^2) - 4*3*Pi/(9*Pi^2 - 4*x^2*A(x)^2) + 4*5*Pi/(25*Pi^2 - 4*x^2*A(x)^2) - 4*7*Pi/(49*Pi^2 - 4*x^2*A(x)^2) +- ...
E.g.f. A(x) is related to the Euler numbers (A000364) by the series:
A(x*cos(x)) = 1/cos(x) = 1 + x^2/2! + 5*x^4/4! + 61*x^6/6! + 1385*x^8/8! + ... + A000364(n)*x^(2*n)/(2*n)! + ...
A(x) = 1 + x^2*A(x)^2/2! + 5*x^4*A(x)^4/4! + 61*x^6*A(x)^6/6! + 1385*x^8*A(x)^8/8! + ... + A000364(n)*x^(2*n)*A(x)^(2*n)/(2*n)! + ...

Maple

S:= series(RootOf(z*cos(x*z)-1, z), x, 101):
seq(coeff(S, x, 2*j)*(2*j)!, j=0..50); # Robert Israel, Nov 24 2015

Mathematica

Table[(CoefficientList[1/x*InverseSeries[Series[x*Cos[x], {x, 0, 43}], x], x]*Range[0, 42]!)[[n]], {n, 1, 41, 2}] (* Vaclav Kotesovec, Jan 23 2014 *)
Table[Sum[(-1)^(m+n) m^(2n) Binomial[2k, k-m] (2n+k)!/(k (2k-2)!! (2n+1)!), {k, 1, 2n}, {m, 1, k}] + KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 24 2015 *)

Prog

(PARI) {a(n)=(2*n)!*polcoeff((1/x)*serreverse(x*cos(x +O(x^(2*n+2)))), 2*n)}
(PARI) /* From Vladimir Kruchinin's formula in A185142: */
{a(n)=if(n==0, 1, 2*sum(k=1, 2*n, binomial(2*n+k, 2*n)*sum(j=1, k, sum(i=0, floor((j-1)/2), (j-2*i)^(2*n)*binomial(j, i))*binomial(k, j)*(-1)^(n-j)/2^j)))/(2*n+1))}

Crossrefs

Cf. A000364, A185142.

Sequence in context: A086265 A156138 A229261 * A262386 A298302 A166188

Adjacent sequences: A196870 A196871 A196872 * A196874 A196875 A196876

Keyword

nonn

Author

Paul D. Hanna, Oct 06 2011

Status

approved