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A003701
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E.g.f. exp(x)/cos(x).
(Formerly M1259)
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4
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1, 1, 2, 4, 12, 36, 152, 624, 3472, 18256, 126752, 814144, 6781632, 51475776, 500231552, 4381112064, 48656756992, 482962852096, 6034272215552, 66942218896384, 929327412759552, 11394877025289216, 174008703107274752
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OFFSET
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0,3
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COMMENTS
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Binomial transform of A000364 (with interpolated zeros). Hankel transform is A055209. [Paul Barry, Jan 12 2009]
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REFERENCES
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T. Chow, Fair permutations and random k-sets, Problem 11523, Amer. Math. Monthly 117 (October 2010), 741; solution by Jim Simons, Amer. Math. Monthly 119 (November 2012), 801-803.
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..100
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
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FORMULA
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G.f.: 1/(1-x-x^2/(1-x-4x^2/(1-x-9x^2/(1-x-16x^2.... (continued fraction). [Paul Barry, Jan 12 2009]
E.g.f.: exp(x)*sec(x). [Zerinvary Lajos, Apr 05 2009]
E.g.f.: 1+x/H(0); H(k)=4k+1-x+x^2*(4k+1)/((2k+1)*(4k+3)-x^2+x*(2k+1)*(4k+3)/(2k+2-x+x*(2k+2)/H(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011
G.f.: 1/G(0) where G(k)= 1 - 2*x*(k+1)/(1 + 1/(1 + 2*x*(k+1)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 20 2012
G.f.: -1/x/Q(0), where Q(k)= 1 - 1/x - (k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 26 2013
G.f.: (1-x)/Q(0), where Q(k)= (1-x)^2 - (1-x)^2*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
a(n) ~ n! * ((-1)^n*exp(-Pi/2) + exp(Pi/2)) *(2/Pi)^(n+1). - Vaclav Kotesovec, Oct 08 2013
G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)/( x*(2*k+1) - 1/(1 + x*(2*k+1)/( x*(2*k+1) + 1/(1 - x*(2*k+2)/( x*(2*k+2) - 1/(1 + x*(2*k+2)/( x*(2*k+2) + 1/Q(k+1) ))))))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
G.f.: Q(0)/(1-x), where Q(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 - (1-x)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013
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EXAMPLE
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1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 152*x^6 + 624*x^7 + 3472*x^8 + ...
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MAPLE
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G(x):= exp(x)*sec(x): f[0]:=G(x): for n from 1 to 54 do f[n]:= diff(f[n-1], x) od: x:=0: seq(f[n], n=0..22); # Zerinvary Lajos, Apr 05 2009
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x ] / Cos[x], {x, 0, n}]] (* Michael Somos, Jun 06 2012 *)
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PROG
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(PARI) x='x+O('x^66); Vec(serlaplace(exp(x)/cos(x))) \\ Joerg Arndt, May 07 2013
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CROSSREFS
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Cf. A062272, A062161.
Bisections are A000795 and A002084(n).
Sequence in context: A046993 A010551 A111942 * A255432 A193049 A114500
Adjacent sequences: A003698 A003699 A003700 * A003702 A003703 A003704
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KEYWORD
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nonn
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AUTHOR
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R. H. Hardin
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EXTENSIONS
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Extended and reformatted 03/97.
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STATUS
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approved
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