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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
J. M. Gandhi, The coefficients of sinh x/ cos x. Canad. Math. Bull. 13 1970 305-310.
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 A365650
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
a(13)-a(15) from Andrew Howroyd, Feb 05 2018
STATUS
approved

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Last modified March 29 03:41 EDT 2024. Contains 371264 sequences. (Running on oeis4.)