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A002084 Sinh(x) / cos(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.

%I A002084 M3667 N1493 #50 Oct 15 2023 01:42:57
%S A002084 1,4,36,624,18256,814144,51475776,4381112064,482962852096,
%T A002084 66942218896384,11394877025289216,2336793875186479104,
%U A002084 568240131312188379136,161669933656307658932224,53204153193639888357113856,20053432927718528320240287744
%N A002084 Sinh(x) / cos(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.
%C A002084 Gandhi proves that a(n) == 1 (mod 2n+1) if 2n+1 is prime, that a(2n+1) == 4 (mod 10), and that a(2n+2) == 6 (mod 10). - _Charles R Greathouse IV_, Oct 16 2012
%D A002084 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002084 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002084 T. D. Noe, <a href="/A002084/b002084.txt">Table of n, a(n) for n = 0..50</a>
%H A002084 J. M. Gandhi, <a href="http://dx.doi.org/10.4153/CMB-1970-059-9">The coefficients of sinh x/ cos x</a>. Canad. Math. Bull. 13 1970 305-310.
%H A002084 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.
%F A002084 E.g.f.: sinh(x)/cos(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.
%F A002084 a(n) = Sum_{k=0..n} binomial(2n+1, 2k+1)*A000364(n-k) = Sum_{k=0..n} A103327(n, k)*A000324(n-k) = Sum_{k=0..n} (-1)^(n-k)*A104033(n, k). - _Philippe Deléham_, Aug 27 2005
%F A002084 a(n) ~ sinh(Pi/2) * 2^(2*n + 3) * (2*n + 1)! / Pi^(2*n+2). - _Vaclav Kotesovec_, Jul 05 2020
%e A002084 x + 2/3*x^3 + 3/10*x^5 + 13/105*x^7 + 163/3240*x^9 + ...
%t A002084 With[{nn=30},Take[CoefficientList[Series[Sinh[x]/Cos[x],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Jul 17 2012 *)
%o A002084 (Sage) # Generalized algorithm of L. Seidel (1877)
%o A002084 def A002084_list(n) :
%o A002084     R = []; A = {-1:0, 0:0}
%o A002084     k = 0; e = 1
%o A002084     for i in range(2*n) :
%o A002084         Am = 1 if e == -1 else 0
%o A002084         A[k + e] = 0
%o A002084         e = -e
%o A002084         for j in (0..i) :
%o A002084             Am += A[k]
%o A002084             A[k] = Am
%o A002084             k += e
%o A002084         if e == 1 : R.append(A[i//2])
%o A002084     return R
%o A002084 A002084_list(10) # _Peter Luschny_, Jun 02 2012
%o A002084 (PARI) a(n)=n++;my(v=Vec(1/cos(x+O(x^(2*n+1)))));v=vector(n,i,v[2*i-1]*(2*i-2)!);sum(g=1,n,binomial(2*n-1,2*g-2)*v[g]) \\ _Charles R Greathouse IV_, Oct 16 2012
%o A002084 (PARI) list(n)=n++;my(v=Vec(1/cos(x+O(x^(2*n+1)))));v=vector(n,i,v[2*i-1]*(2*i-2)!);vector(n,k,sum(g=1,k,binomial(2*k-1,2*g-2)*v[g])) \\ _Charles R Greathouse IV_, Oct 16 2012
%Y A002084 Cf. A002085.
%K A002084 nonn,easy
%O A002084 0,2
%A A002084 _N. J. A. Sloane_
%E A002084 a(13)-a(15) from _Andrew Howroyd_, Feb 05 2018