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A122045
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Euler (or secant) numbers E(n).
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20
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1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765, 0, -199360981, 0, 19391512145, 0, -2404879675441, 0, 370371188237525, 0, -69348874393137901, 0, 15514534163557086905, 0, -4087072509293123892361, 0, 1252259641403629865468285, 0, -441543893249023104553682821
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OFFSET
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0,5
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COMMENTS
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The convention in the OEIS is that the alternate zeros are normally omitted in such sequences. See A000364 for the official version of this sequence.
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LINKS
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Table of n, a(n) for n=0..30.
T. J. Stieltjes, Sur la reduction en fraction continue d'une serie procedant suivant les puissances descendantes d'une variable, Ann. Fac. Sc. Toulouse 3 (1889) 1-17
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FORMULA
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G.f.: 1/U(0) where U(k)= 1 - x + x*(k+1)/(1 - x*(k+1)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 14 2012
G.f.: 1/U(0) where U(k)= 1 - x^2 + x^2*(2*k+1)*(2*k+2)/(1 + x^2*(2*k+1)*(2*k+2)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 15 2012
E.g.f.: (1-x)/U(0) where U(k)= 1 - x/(1 - x/(x - (2*k+1)*(2*k+2)/U(k+1)); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 17 2012
E.g.f.: 1 - x^2/U(0) where U(k)= (2*k+1)*(2*k+2) + x^2 - x^2*(2*k+1)*(2*k+2)/U(k+1) ; (continued fraction, Euler's 1=st kind, 1-step). - Sergei N. Gladkovskii, Oct 19 2012
E.g.f.: 1/U(0) where U(k)= 1 + x^2/(2*(2*k+1)*(4*k+1) - 2*x^2*(2*k+1)*(4*k+1)/(x^2 + 4*(4*k+3)*(k+1)/U(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Oct 21 2012
E.g.f.: (2 + x^4/(U(0)*(x^2-2) - 2))/(2-x^2) where U(k)= 4*k+4 + 1/(1 + x^2/(2 - x^2 + (2*k+3)*(2*k+4)/U(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Oct 28 2012
G.f.: 1/G(0) where G(k) = 1 + x^2*(2*k+1)^2/(1 + x^2*(2*k+2)^2/G(k+1) ); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: 1/S(0) where S(k) = 1 + x^2*(k+1)^2/S(k+1); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: 1 - x/(1+x) + x/(1+x)/Q(0), where Q(k)= 1 - x + x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: -1/x/Q(0), where Q(k)= -1/x + (k+1)^2/Q(k+1); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Apr 25 2013
G.f.: 1/(1-x)/Q(0) + 1/(1-x), where Q(k)= 1 - 1/x + (k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 26 2013
G.f.: x/(x-1)/Q(0) + 1/(1-x), where Q(k)= 1 - x + x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 27 2013
G.f.: 1-x/(1+x) + x/(1+x)/Q(0), where Q(k)= 1 + x + (k+1)*(k+2)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
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MAPLE
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seq(euler(n) , n=0..31); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 15 2009]
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MATHEMATICA
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Table[EulerE[n], {n, 0, 30}]
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PROG
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(Sage) [euler_number(i) for i in range(31)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 15 2009]
(PARI) a(n)=sumalt(k=0, (-1)^k*(2*k+1)^n)*2 /* Gottfried Helms, Mar 09 2012 */
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CROSSREFS
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Cf. A000364, A028296.
Sequence in context: A186746 A208928 A103709 * A073911 A157302 A222327
Adjacent sequences: A122042 A122043 A122044 * A122046 A122047 A122048
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KEYWORD
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sign
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AUTHOR
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Roger Bagula, Sep 13 2006
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EXTENSIONS
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Edited by N. J. A. Sloane, Sep 17 2006
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STATUS
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approved
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