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A122045 Euler (or secant) numbers E(n). 33
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

T. J. Stieltjes, Sur la réduction en fraction continue d'une série procédant suivant les puissances descendantes d'une variable, Ann. Fac. Sc. Toulouse 3 (1889) 1-17.

FORMULA

E.g.f.: sech(x). - Michael Somos, Mar 11 2014

a(n) = Sum_{k>=0} (-1)^k*(2*k+1)^n)*2. - Gottfried Helms, Mar 09 2012

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

E.g.f.: 1 - T(0)*x^2/(2+x^2), where T(k) = 1 - x^2*(2*k+1)*(2*k+2)/(x^2*(2*k+1)*(2*k+2) - ((2*k+1)*(2*k+2) + x^2)*((2*k+3)*(2*k+4) + x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 13 2013

G.f.: T(0), where T(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 + 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 13 2013

EXAMPLE

G.f. = 1 - x^2 + 5*x^4 - 61*x^6 + 1385*x^8 - 50521*x^10 + 2702765*x^12 + ...

MAPLE

seq(euler(n) , n=0..31); # Zerinvary Lajos, Mar 15 2009

P := proc(n, x) option remember; if n = 0 then 1 else

   (n*x+(1/2)*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x);

   expand(%) fi end:

A122045 := n -> (-1)^n*subs(x=-1, P(n, x)):

seq(A122045(n), n=0..30);  # Peter Luschny, Mar 07 2014

MATHEMATICA

Table[EulerE[n], {n, 0, 30}]

PROG

(Sage) [euler_number(i) for i in range(31)] # Zerinvary Lajos, Mar 15 2009

(PARI) x='x+O('x^66); Vec(serlaplace(1/cosh(x))) \\ Joerg Arndt, Mar 10 2014

CROSSREFS

Cf. A000364, A028296.

Sequence in context: A186746 A208928 A103709 * A073911 A157302 A222327

Adjacent sequences:  A122042 A122043 A122044 * A122046 A122047 A122048

KEYWORD

sign

AUTHOR

Roger L. Bagula, Sep 13 2006

EXTENSIONS

Edited by N. J. A. Sloane, Sep 17 2006

STATUS

approved

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Last modified October 26 01:20 EDT 2014. Contains 248566 sequences.