A046977 -id:A046977 - cp's OEIS frontend

A046976 Numerators of Taylor series for sec(x) = 1/cos(x).

1, 1, 5, 61, 277, 50521, 540553, 199360981, 3878302429, 2404879675441, 14814847529501, 69348874393137901, 238685140977801337, 4087072509293123892361, 13181680435827682794403, 441543893249023104553682821, 2088463430347521052196056349

Offset: 0

N. J. A. Sloane

Also numerator of beta(2n+1)/Pi^(2n+1), where beta(m) = Sum_{k>=0} (-1)^k/(2k+1)^m.

			sec(x) = 1 + (1/2)*x^2 + (5/24)*x^4 + (61/720)*x^6 + (277/8064)*x^8 + (50521/3628800)*x^10 + ...

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 384, Problem 15.
G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

Seiichi Manyama, Table of n, a(n) for n = 0..243 (terms 0..100 from T. D. Noe)
Xuming Chen, Recursive formulas for zeta(2*k) and L(2*k-1), Coll. Math. J. 26 (5) (1995) 372-376. See numerators of D_(2k-1).
Jan W. H. Swanepoel, A Short Simple Probabilistic Proof of a Well Known Identity and the Derivation of Related New Identities Involving the Bernoulli Numbers and the Euler Numbers, Integers (2025) Vol. 25, Art. No. A50. See p. 4.
Eric Weisstein's World of Mathematics, Secant
Eric Weisstein's World of Mathematics, Dirichlet Beta Function
Eric Weisstein's World of Mathematics, Hyperbolic Secant

Cf. A000364, A046977, A053005, A099612.

ZBS := z -> (Zeta(0,z,1/4) - Zeta(0,z,3/4))/(2^z-2):
R := n -> (-1)^floor(n/2)*(2^n-4^n)*ZBS(1-n)/(n-1)!:
seq(numer(R(2*n+1)), n=0..16); # Peter Luschny, Aug 25 2015

Numerator[Partition[CoefficientList[Series[Sec[x], {x, 0, 30}], x], 2][[All,1]]]

a(n)/A046977(n) = A000364(n)/(2n)!.

Let ZBS(z) = (HurwitzZeta(z,1/4) - HurwitzZeta(z,3/4))/(2^z-2) and R(z) = (cos(z*Pi/2)+sin(z*Pi/2))*(2^z-4^z)*ZBS(1-z)/(z-1)!. Then a(n) = numerator(R(2*n+1)) and A046977(n) = denominator(R(2*n+1)). - Peter Luschny, Aug 25 2015

A162445 A sequence related to the Beta function.

Original entry on oeis.org

1, 8, 384, 46080, 2064384, 3715891200, 392398110720, 1428329123020800, 274239191619993600, 1678343852714360832000, 102043306245033138585600, 4714400748520531002654720000, 160144566965128191597871104000

Offset: 0

Johannes W. Meijer, Jul 06 2009

We define F(z) = Beta(1/2-z/2,1/2+z/2)/Beta(1/2,1/2) = 1/sin(Pi*(1+z)/2) with Beta(z,w) the Beta function. See A008956 for a closely related function.
For the Taylor series expansion of F(z) we can write F(z) = sum(b(n)*(Pi*z)^(2*n)/a(n), n=0..infinity) with b(n) = A046976(n) and a(n) the sequence given above.
We can also write F(z) = sum(c(n)*(Pi*z)^(2*n)/d(n), n=0..infinity) with c(n) = A000364(n) and d(n) = A067624(n).
If p(n) is the exponent of the prime factor 2 in a(n) than p(n) = A120738(n) and 2^p(n) = A061549(n) = abs((4*n)!!/A117972(n)).

Bisection of A050971
Equals 2^(2*n)*A046977(n)
Cf. A008956, A046976, A000364, A067624, A120738, A061549 and A117972.

Denominator[Table[EulerE[2n]/(4n)!!,{n,0,20}]] (* Harvey P. Dale, Jun 23 2013 *)

a(n) = denom(euler(2*n)/(4*n)!!)

cp's OEIS Frontend

A046976 Numerators of Taylor series for sec(x) = 1/cos(x).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula