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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A122045 Euler (or secant) numbers E(n).

Original entry on oeis.org

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
Offset: 0

Views

Author

Roger L. Bagula, Sep 13 2006

Keywords

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

Examples

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

References

  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 5, page 41.

Links

Crossrefs

Cf. A000364, A002144, A028296, A265237, A331978, A350972.

Programs

  • 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
  • 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
ptan := proc(n) option remember; if irem(n, 2) = 1 then 0 else
    -add(`if`(k=0, 1, binomial(n, k)*ptan(n - k)), k = 0..n-1,2) fi end:
A122045 := n -> ifelse(n = 0, 1, ptan(n)):
seq(A122045(n), n = 0..30); # Peter Luschny, Jun 06 2022
  • 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 *)
a[0] := 1; a[n_?OddQ] := 0; a[n_?EvenQ] := a[n] = -Sum[a[k] Binomial[n, k], {k, 0, n - 1, 2}]; Map[a, Range[0, 30]] (* Oliver Seipel, May 20 2024 *)
  • Maxima
    a[n]:=if n<2 then 1-n else sum(-a[n-2*k]*binomial(n,2*k),k,1,floor(n/2));
makelist(a[n],n,0,50); /* Tani Akinari, Sep 15 2023 */
  • 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
  • Python
    from functools import cache
from math import comb as binomial
@cache
def ptan(n):   # see also A331978 and A350972.
    return (0 if n % 2 == 1 else
    -sum(binomial(n,k) * ptan(n-k) if k > 0 else 1 for k in range(0, n-1, 2)))
def A122045(n): return 1 if n == 0 else ptan(n)
print([A122045(n) for n in range(31)])  # Peter Luschny, Jun 06 2022
  • Sage
    [euler_number(i) for i in range(31)] # Zerinvary Lajos, Mar 15 2009

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)*((n+1) mod 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
For n > 1, a(n) = -Sum_{k=1..floor(n/2)} a(n-2*k)*binomial(n,2*k). - Tani Akinari, Sep 15 2023

Extensions

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

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