The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A002084 Sinh x / cos x = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!. (Formerly M3667 N1493) 15
 1, 4, 36, 624, 18256, 814144, 51475776, 4381112064, 482962852096, 66942218896384, 11394877025289216, 2336793875186479104, 568240131312188379136, 161669933656307658932224, 53204153193639888357113856, 20053432927718528320240287744 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..50 J. M. Gandhi, The coefficients of sinh x/ cos x. Canad. Math. Bull. 13 1970 305-310. Peter Luschny, An old operation on sequences: the Seidel transform FORMULA E.g.f.: sinh(x)/cos(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!. 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 a(n) ~ sinh(Pi/2) * 2^(2*n + 3) * (2*n + 1)! / Pi^(2*n+2). - Vaclav Kotesovec, Jul 05 2020 EXAMPLE x + 2/3*x^3 + 3/10*x^5 + 13/105*x^7 + 163/3240*x^9 + ... MATHEMATICA 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 *) PROG (Sage) # Generalized algorithm of L. Seidel (1877) def A002084_list(n) :     R = []; A = {-1:0, 0:0}     k = 0; e = 1     for i in range(2*n) :         Am = 1 if e == -1 else 0         A[k + e] = 0         e = -e         for j in (0..i) :             Am += A[k]             A[k] = Am             k += e         if e == 1 : R.append(A[i//2])     return R A002084_list(10) # Peter Luschny, Jun 02 2012 (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 (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 CROSSREFS Cf. A002085. Sequence in context: A263445 A241029 A002761 * A135867 A268470 A214347 Adjacent sequences:  A002081 A002082 A002083 * A002085 A002086 A002087 KEYWORD nonn,easy AUTHOR EXTENSIONS a(13)-a(15) from Andrew Howroyd, Feb 05 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 26 13:24 EDT 2022. Contains 354883 sequences. (Running on oeis4.)