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

%I A212435 #53 Feb 21 2024 10:54:58
%S A212435 1,-1,-3,11,57,-361,-2763,24611,250737,-2873041,-36581523,512343611,
%T A212435 7828053417,-129570724921,-2309644635483,44110959165011,
%U A212435 898621108880097,-19450718635716001,-445777636063460643,10784052561125704811,274613643571568682777
%N A212435 Expansion of e.g.f.: exp(-x) / cosh(2*x).
%H A212435 Vincenzo Librandi, <a href="/A212435/b212435.txt">Table of n, a(n) for n = 0..200</a>
%F A212435 E.g.f.: 2 * exp(x) / (exp(4*x) + 1).
%F A212435 E.g.f. is the reciprocal of the e.g.f. of A046717.
%F A212435 a(n) = (-1)^n * A188458(n) = (-1)^floor((n + 1) / 2) * A001586(n).
%F A212435 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
%F A212435 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
%F A212435 a(n) ~ n! * (cos(Pi*n/2)-sin(Pi*n/2)) * 2^(2*n+3/2) / Pi^(n+1). - _Vaclav Kotesovec_, Feb 25 2014
%F A212435 From _Peter Bala_, Mar 10 2015: (Start)
%F A212435 a(n) = 4^n*E(n,1/4).
%F A212435 O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - x*(4*k + 1)).
%F A212435 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)
%e A212435 G.f. = 1 - x - 3*x^2 + 11*x^3 + 57*x^4 - 361*x^5 - 2763*x^6 + 24611*x^7 + ...
%t A212435 CoefficientList[Series[2*E^x/(E^(4*x)+1), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 25 2014 *)
%t A212435 a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ -x] / Cosh[ 2 x], {x, 0, n}]]; (* _Michael Somos_, Aug 26 2015 *)
%o A212435 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( exp(-x + A) / cosh( 2*x + A), n))};
%o A212435 (Sage)
%o A212435 @CachedFunction
%o A212435 def p(n,x) :
%o A212435     if n == 0 : return 1
%o A212435     w = -1 if n%2 == 0 else  0
%o A212435     v =  1 if n%2 == 0 else -1
%o A212435     return v*add(p(k,0)*binomial(n,k)*(x^(n-k)+w) for k in range(n)[::2])
%o A212435 def A212435(n) : return 2^n*p(n, 1/2)
%o A212435 [A212435(n) for n in (0..20)]  # _Peter Luschny_, Jul 19 2012
%o A212435 (Magma) m:=30; R<x>:=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
%Y A212435 Cf. A001586, A046717, A188458, A188514, A255883.
%K A212435 sign
%O A212435 0,3
%A A212435 _Michael Somos_, Jun 21 2012