A046976 -id:A046976 - cp's OEIS frontend

A050970 Numerator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} (4k+1)^(-n).

1, 1, 1, 1, 5, 1, 61, 17, 277, 31, 50521, 691, 540553, 5461, 199360981, 929569, 3878302429, 3202291, 2404879675441, 221930581, 14814847529501, 4722116521, 69348874393137901, 56963745931, 238685140977801337, 14717667114151

Offset: 1

Eric W. Weisstein

Reduced numerators of Favard constants.

			The first few values of S(n)/Pi^n are 1/4, 1/8, 1/32, 1/96, 5/1536, 1/960, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..200
N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001.
N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
Z. K. Silagadze, Comment on the sums S(n) = sum(k=-inf..inf) 1/(4k+1)^n, (2012) arXiv:1207.2055
Eric Weisstein's World of Mathematics, Favard Constants

Denominators: A068205. See also A050971.

Cf. A000828, A046982.

S := proc(n, k) option remember; if k = 0 then `if`(n = 0, 1, 0) else
S(n, k - 1) + S(n - 1, n - k) fi end: EZ := n -> S(n, n)/(2^n * n!):
A050970 := n -> numer(EZ(n-1)): seq(A050970(n), n=1..26); # Peter Luschny, Aug 02 2017
# alternative
A050970 := proc(n)
    if type(n,'even') then
        (-1)^(n/2)*2^(n-2)/(n-1)!*euler(n-1,0) ;
    else
        (-1)^((n-1)/2)*2^(n-2)/(n-1)!*euler(n-1,1/2) ;
    end if;
    %/2^n ;
    numer(%) ;
end proc:
seq(A050970(n),n=1..20) ; # R. J. Mathar, Jun 26 2024

s[n_] := Sum[(4*k + 1)^(-n), {k, -Infinity, Infinity}]; a[n_] := Numerator[FullSimplify[s[n]/Pi^n]]; a[1] = 1; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Oct 25 2012 *)
s[n_?EvenQ] := (-1)^(n/2-1)*(2^n-1)*BernoulliB[n]/(2*n!); s[n_?OddQ] := (-1)^((n-1)/2)*2^(-n-1)*EulerE[n-1]/(n-1)!; Table[s[n] // Numerator, {n, 1, 26}] (* Jean-François Alcover, May 13 2013 *)
a[n_] := 4*Sum[((-1)^k/(2*k+1))^n, {k, 0, Infinity}] /. Pi -> 1 // Numerator; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Jun 20 2014 *)
Table[4/(2 Pi)^n LerchPhi[(-1)^n, n, 1/2], {n, 21}] // Numerator (* Eric W. Weisstein, Aug 02 2017 *)
Table[4/Pi^n If[Mod[n, 2] == 0, DirichletLambda, DirichletBeta][n], {n, 21}] // Numerator (* Eric W. Weisstein, Aug 02 2017 *)

{a(n) = if( n<0, 0, numerator( polcoeff( 1 / (1 - tan(x/4 + x * O(x^n))), n)))}; /* Michael Somos, Nov 11 2014 */

There is a simple formula in terms of Euler and Bernoulli numbers.
a(2n) = A046976(n), a(2n+1) = A089171(n+1) (conjectured).
Numerator of coefficients of expansion of (sec(x/2) + tan(x/2) + 1)/2 in powers of x. - Sergei N. Gladkovskii, Nov 11 2014

Entry revised by N. J. A. Sloane, Mar 24 2002

A046977 Denominators of Taylor series for sec(x). Also denominators of Taylor series for sech(x) = 1/cosh(x).

Original entry on oeis.org

1, 2, 24, 720, 8064, 3628800, 95800320, 87178291200, 4184557977600, 6402373705728000, 97316080327065600, 1124000727777607680000, 9545360026665222144000, 403291461126605635584000000, 3209350995912777478963200000, 265252859812191058636308480000000

Offset: 0

N. J. A. Sloane

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

G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

Seiichi Manyama, Table of n, a(n) for n = 0..225 (terms 0..100 from T. D. Noe)
Eric Weisstein's World of Mathematics, Secant
Eric Weisstein's World of Mathematics, Hyperbolic Secant

Cf. A000364, A046976, 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(denom(R(2*n+1)), n=0..16); # Peter Luschny, Aug 25 2015

Table[ EulerE[n]/n! // Denominator, {n, 0, 30, 2}] (* Jean-François Alcover, Oct 04 2012 *)

A046976(n)/a(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) = denominator(R(2*n+1)) and A046976(n) = numerator(R(2*n+1)). - Peter Luschny, Aug 25 2015

A053005 Denominator of beta(2n+1)/Pi^(2n+1), where beta(m) = Sum_{k=0..inf} (-1)^k/(2k+1)^m.

Original entry on oeis.org

4, 32, 1536, 184320, 8257536, 14863564800, 1569592442880, 5713316492083200, 1096956766479974400, 6713375410857443328000, 408173224980132554342400, 18857602994082124010618880000, 640578267860512766391484416000

Offset: 0

N. J. A. Sloane, Feb 21 2000

			beta(5) = 5*Pi^5/1536 so a(2)=1536.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 384, Problem 15.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Problem 37, beta(n).

T. D. Noe, Table of n, a(n) for n = 0..100
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, Dirichlet Beta Function

Cf. A046976.

beta[1] = Pi/4; beta[m_] := (Zeta[m, 1/4] - Zeta[m, 3/4])/4^m; a[n_, p_] := a[n, p] = beta[2*n+1]/Pi^(2*n+1) // N[#, p]& // Rationalize[#, 0]& // Denominator; a[n_] := Module[{p = 16}, a[n, p]; p = 2*p; While[a[n, p] != a[n, p/2], p = 2*p]; a[n, p]]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Aug 19 2013 *)

A273192 Unsigned numerators of the nonzero coefficients in the expansion of 1/hypergeom([], [Seq_{k=1..m-1} k/m], (x/m)^m) for m = 4.

Original entry on oeis.org

1, 1, 23, 33661, 20125603, 288294050521, 335455351400261, 60921822444067346581, 42334470849679755333739, 125339574046311949415000577841, 3763027389086366356144399489191943

Offset: 0

Peter Luschny, Jun 06 2016

Cf. A000012 (m=1), A046976 (m=2), A273193, A273194.

Hlist := proc(m, size) local H, S;
H := m -> hypergeom([], [seq(k/m, k=1..m-1)], (x/m)^m);
S := m -> series(1/H(m), x, (m+1)*size);
seq((-1)^n*numer(coeff(S(m), x, m*n)), n=0..size) end:
A273192_list := size -> Hlist(4, size);

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

A050970 Numerator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} (4k+1)^(-n).

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A046977 Denominators of Taylor series for sec(x). Also denominators of Taylor series for sech(x) = 1/cosh(x).

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A053005 Denominator of beta(2n+1)/Pi^(2n+1), where beta(m) = Sum_{k=0..inf} (-1)^k/(2k+1)^m.

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A273192 Unsigned numerators of the nonzero coefficients in the expansion of 1/hypergeom([], [Seq_{k=1..m-1} k/m], (x/m)^m) for m = 4.

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A162445 A sequence related to the Beta function.

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