A212435 Expansion of e.g.f.: exp(-x) / cosh(2*x).

1, -1, -3, 11, 57, -361, -2763, 24611, 250737, -2873041, -36581523, 512343611, 7828053417, -129570724921, -2309644635483, 44110959165011, 898621108880097, -19450718635716001, -445777636063460643, 10784052561125704811, 274613643571568682777

Offset: 0

Author

Michael Somos, Jun 21 2012

Keywords

• sign

Examples

			G.f. = 1 - x - 3*x^2 + 11*x^3 + 57*x^4 - 361*x^5 - 2763*x^6 + 24611*x^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A001586, A046717, A188458, A188514, A255883.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x)/Cosh(2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 10 2018

Mathematica

CoefficientList[Series[2*E^x/(E^(4*x)+1), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 25 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ -x] / Cosh[ 2 x], {x, 0, n}]]; (* Michael Somos, Aug 26 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( exp(-x + A) / cosh( 2*x + A), n))};

Sage

@CachedFunction
def p(n,x) :
    if n == 0 : return 1
    w = -1 if n%2 == 0 else  0
    v =  1 if n%2 == 0 else -1
    return v*add(p(k,0)*binomial(n,k)*(x^(n-k)+w) for k in range(n)[::2])
def A212435(n) : return 2^n*p(n, 1/2)
[A212435(n) for n in (0..20)]  # Peter Luschny, Jul 19 2012

Formula

E.g.f.: 2 * exp(x) / (exp(4*x) + 1).

E.g.f. is the reciprocal of the e.g.f. of A046717.

a(n) = (-1)^n * A188458(n) = (-1)^floor((n + 1) / 2) * A001586(n).

E.g.f.: 2/E(0), where E(k) = 1 + (-1)^k/(3^k - 3*9^k*x/(3*3^k*x + (-1)^k*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013

G.f.: conjecture T(0)/(1+x), where T(k) = 1 - 4*x^2*(k+1)^2/(4*x^2*(k+1)^2 + (1+ x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013

a(n) ~ n! * (cos(Pi*n/2)-sin(Pi*n/2)) * 2^(2*n+3/2) / Pi^(n+1). - Vaclav Kotesovec, Feb 25 2014

From Peter Bala, Mar 10 2015: (Start)

a(n) = 4^n*E(n,1/4).

O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - x*(4*k + 1)).

The series expansion exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 - x - x^2 + 5*x^3 + 11*x^4 - 91*x^5 - 391*x^6 + ... appears to have integer coefficients. Cf. A188514, A255883. (End)