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 A122045 Euler (or secant) numbers E(n). 65
 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. Odd primes p such that p | E(p-1) are primes p == 1 (mod 4), A002144. Conjecture: odd composites m such that m | E(m-1) are Carmichael numbers m such that p == 1 (mod 4) for every prime p|m, A265237. - Thomas Ordowski, Feb 06 2020 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Fernando Barbero G., Juan Margalef-Bentabol and Eduardo J.S. Villaseñor, A two-sided Faulhaber-like formula involving Bernoulli polynomials, arXiv:2002.00550 [math.NT], 2020. Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 445-463. Michael D. Hirschhorn, Binomial Identities and Congruences for Euler Numbers, Fibonacci Quart. 53 (2015), no. 4, 319-322. Guodong Liu, Generating functions and generalized Euler numbers, Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 2 (2008), 29-34. See p 32. F. Luca, A. Pizarro-Madariaga and C. Pomerance, On the counting function of irregular primes, 2014. A. Niedermaier and J. Remmel, Analogues of up-down permutations for colored permutations, J. Int. Seq. 13 (2010) 10.5.6. 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 From Sergei N. Gladkovskii, Oct 14 2012 - Oct 13 2013: (Start) Continued fractions: G.f.: 1/U(0) where U(k) = 1 - x + x*(k+1)/(1 - x*(k+1)/U(k+1)). 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)). E.g.f.: (1-x)/U(0) where U(k) = 1 - x/(1 - x/(x - (2*k+1)*(2*k+2)/U(k+1)). 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). 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))). 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))). 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)) (due to T. J. Stieltjes). G.f.: 1/S(0) where S(k) = 1 + x^2*(k+1)^2/S(k+1) (due to T. J. Stieltjes). 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)). G.f.: -1/x/Q(0) where Q(k) = -1/x + (k+1)^2/Q(k+1) (due to T. J. Stieltjes). G.f.: 1/(1-x)/Q(0) + 1/(1-x) where Q(k) = 1 - 1/x + (k+1)*(k+2)/Q(k+1). G.f.: x/(x-1)/Q(0) + 1/(1-x) where Q(k) = 1 - x + x^2*(k+1)*(k+2)/Q(k+1). 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). 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)). G.f.: T(0) where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 + 1/T(k+1)). (End) a(n) = 2^(2*n+1)*(zeta(-n,1/4) - zeta(-n,3/4)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015 a(n) = 2^n*(2^(n+1)/(n+1))*(B(n+1, 3/4) - B(n+1, 1/4)) where B(n,x) is the n-th Bernoulli polynomial. See Liu link. - Michel Marcus, May 20 2017 [This is the same as: a(n) = -4^(n+1)*B(n+1, 1/4)*mod(n+1, 2)/(n+1). Peter Luschny, Oct 30 2020] a(n) = 2*Im(PolyLog(-n, I)). - Peter Luschny, Sep 29 2020 a(4n) == 5 (mod 60) and a(4n+2) == -1 (mod 60). See Hirschhorn. - Michel Marcus, Jan 11 2022 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}] Range[0, 30]! CoefficientList[ Series[ Sech[x], {x, 0, 30}], x] (* Robert G. Wilson v, Aug 08 2018 *) 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 (PARI) a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1); \\ Michel Marcus, May 20 2017 (Python) from sympy import bernoulli as B def a(n): return int(2**n*2**(n + 1)*(B(n + 1, 3/4) - B(n + 1, 1/4))/(n + 1)) print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017, after PARI code by Michel Marcus (MAGMA) m:=35; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/Cosh(x) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 13 2020 CROSSREFS Cf. A000364, A002144, A028296, A265237. Sequence in context: A348209 A297206 A103709 * A294314 A347599 A266324 Adjacent sequences:  A122042 A122043 A122044 * A122046 A122047 A122048 KEYWORD sign,changed 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 January 22 07:31 EST 2022. Contains 350481 sequences. (Running on oeis4.)