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A002084
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Sinh x / cos x = sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.
(Formerly M3667 N1493)
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9
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1, 4, 36, 624, 18256, 814144, 51475776, 4381112064, 482962852096, 66942218896384, 11394877025289216, 2336793875186479104, 568240131312188379136
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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J. M. Gandhi, The coefficients of sinh x/ cos x. Canad. Math. Bull. 13 1970 305-310.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..50
Peter Luschny, An old operation on sequences: the Seidel transform
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FORMULA
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E.g.f.: sinh(x)/cos(x) = sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.
a(n) = Sum_{k, 0<=k<=n} binomial(2n+1, 2k+1)*A000364(n-k) = Sum_{k, 0<=k<=n} A103327(n, k)*A000324(n-k) = Sum_{k, 0<=k<=n} (-1)^(n-k)*A104033(n, k) . - Philippe Deléham, Aug 27 2005
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EXAMPLE
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x + 2/3*x^3 + 3/10*x^5 + 13/105*x^7 + 163/3240*x^9 + ...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A002085.
Sequence in context: A086879 A241029 A002761 * A135867 A214347 A208732
Adjacent sequences: A002081 A002082 A002083 * A002085 A002086 A002087
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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