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A046990
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Numerators of Taylor series for log(1/cos(x)). Also from log(cos(x)).
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10
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0, 1, 1, 1, 17, 31, 691, 10922, 929569, 3202291, 221930581, 9444233042, 56963745931, 29435334228302, 2093660879252671, 344502690252804724, 129848163681107301953, 868320396104950823611, 209390615747646519456961, 28259319101491102261334882
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OFFSET
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0,5
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..100
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FORMULA
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Let q(n) = Sum_{k=0..n-1} (-1)^k*A201637(n-1,k) then a(n) = numerator((-1)^(n-1)*q(2*n)/(2*n)!). - Peter Luschny, Nov 16 2012
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EXAMPLE
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log(1/cos(x)) = 1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...
log(cos(x)) = -(1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...).
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MAPLE
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q:= proc(n) add((-1)^k*combinat[eulerian1](n-1, k), k=0..n-1) end: A046990:= n -> numer((-1)^(n-1)*q(2*n)/(2*n)!):
seq(A046990(n), n=0..19); # Peter Luschny, Nov 16 2012
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MATHEMATICA
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Join[{0}, Numerator[Select[CoefficientList[Series[Log[1/Cos[x]], {x, 0, 40}], x], #!=0&]]] (* Harvey P. Dale, Jul 27 2011 *)
a[n_] := Numerator[((-4)^n-(-16)^n)*BernoulliB[2*n]/2/n/(2*n)!]; a[0] = 0; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 11 2014, after Charles R Greathouse IV *)
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PROG
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(Sage) # uses[eulerian1 from A173018]
def A046990(n):
def q(n):
return add((-1)^k*eulerian1(n-1, k) for k in (0..n-1))
return ((-1)^(n-1)*q(2*n)/factorial(2*n)).numer()
[A046990(n) for n in (0..19)] # Peter Luschny, Nov 16 2012
(PARI) a(n)=numerator(((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!) \\ Charles R Greathouse IV, Nov 06 2013
(PARI) {a(n) = if( n<1, 0, my(m = 2*n); numerator( polcoeff( -log(cos(x + x * O(x^m))), m)))}; /* Michael Somos, Jun 03 2019 */
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CROSSREFS
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Cf. A046991, A002430, A050970.
Sequence in context: A276592 A002425 A275994 * A059212 A335360 A058899
Adjacent sequences: A046987 A046988 A046989 * A046991 A046992 A046993
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KEYWORD
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nonn,easy,frac,nice
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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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