A046982 -id:A046982 - 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

A046983 Denominators of Taylor series for tan(x + Pi/4).

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 45, 315, 63, 2835, 14175, 155925, 93555, 6081075, 42567525, 638512875, 127702575, 10854718875, 97692469875, 1856156927625, 371231385525, 194896477400625, 2143861251406875, 2900518163668125, 2275791174570375

Offset: 0

N. J. A. Sloane

			1 + 2*x + 2*x^2 + (8/3)*x^3 + (10/3)*x^4 + (64/15)*x^5 + (244/45)*x^6 + ...

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

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

Cf. A046982.

A046983 := proc(n)
    coeftayl(tan(x+Pi/4),x=0,n) ;
    denom(%) ;
end proc: # R. J. Mathar, Jan 22 2017

nmax = 24; t[0, 1] = 1; t[0, ] = 0; t[n, k_] := t[n, k] = (k-1)*t[n-1, k-1] + (k+1)*t[n-1, k+1]; Denominator[ Table[ Sum[ t[n, k]/n!, {k, 0, n+1}], {n, 0, nmax} ]] (* Jean-François Alcover, Nov 09 2011 *)
CoefficientList[Series[Tan[x+Pi/4],{x,0,30}],x]//Denominator (* Harvey P. Dale, Sep 05 2023 *)

A360945 a(n) = numerator of (Zeta(2n+1,1/4) - Zeta(2n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

Original entry on oeis.org

1, 2, 10, 244, 554, 202084, 2162212, 1594887848, 7756604858, 9619518701764, 59259390118004, 554790995145103208, 954740563911205348, 32696580074344991138888, 105453443486621462355224, 7064702291984369672858925136, 4176926860695042104392112698

Offset: 0

Artur Jasinski, Feb 26 2023

The function (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) is rational for every positive integer n.
For denominators see A360966.
(Zeta(2*n+1,1/4) + Zeta(2*n+1,3/4))/Zeta(2*n+1) = 4*16^n - 2*4^n; see A193475.
For numerators of the function (Zeta(2*n,1/4) + Zeta(2*n,3/4))/Pi^(2*n) see A361007.
For denominators of the function (Zeta(2*n,1/4) + Zeta(2*n,3/4))/Pi^(2*n) see A036279.
(Zeta(2*n,1/4) - Zeta(2*n,3/4))/beta(2*n) = 16^n (see A001025) where beta is the Dirichlet beta function.
From the above formulas we can express Zeta(k,1/4) and Zeta(k,3/4) for every positive integer k.

			a(0) = 1 because lim_{n->0} (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) = 1.
a(3) = 244 because (Zeta(2*3+1,1/4) - Zeta(2*3+1,3/4))/Pi^(2*3+1) = 244/45.

Cf. A000364, A046982, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173982, A173983, A173984, A173987, A360966, A361007, A361007.

Table[(Zeta[2*n + 1, 1/4] - Zeta[2*n + 1, 3/4]) / Pi^(2*n + 1), {n, 1, 25}] // FunctionExpand // Numerator (* Vaclav Kotesovec, Feb 27 2023 *)
t[0, 1] = 1; t[0, _] = 0;
t[n_, k_] := t[n, k] = (k-1) t[n-1, k-1] + (k+1) t[n-1, k+1];
a[n_] := Sum[t[2n, k]/(2n)!, {k, 0, 2n+1}] // Numerator;
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 15 2023 *)
a[n_] := SeriesCoefficient[Tan[x+Pi/4], {x, 0, 2n}] // Numerator;
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 15 2023 *)

a(n) = numerator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023

a(n) = A046982(2*n).

(Zeta(2*n + 1, 1/4) - Zeta(2*n + 1, 3/4))/(Pi^(2*n + 1)) = A000364(n)*(2*n + 1)*2^(2*n)/(2*n + 1)!.

A360966 a(n) = denominator of (Zeta(2n+1,1/4) - Zeta(2n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

Original entry on oeis.org

1, 1, 3, 45, 63, 14175, 93555, 42567525, 127702575, 97692469875, 371231385525, 2143861251406875, 2275791174570375, 48076088562799171875, 95646113035463615625, 3952575621190533915703125, 1441527579493018251609375, 68739242628124575327993046875, 333120945043988326589504765625

Offset: 0

Artur Jasinski, Apr 09 2023

The function (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) is rational for every positive integer n.

For numerators see A360945.

			a(0) = 1 because lim_{n->0} (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) = 1.
a(3) = 45 because (Zeta(2*3+1,1/4) - Zeta(2*3+1,3/4))/Pi^(2*3+1) = 244/45.

Wikipedia, Hurwitz zeta function

Bisection of A046983.
Cf. A360945 (numerators).
Cf. A000364, A046982, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173982, A173983, A173984, A173987, A361007.

Table[(Zeta[2*n + 1, 1/4] - Zeta[2*n + 1, 3/4]) / Pi^(2*n + 1), {n, 0, 25}] // FunctionExpand // Denominator
(* Second program: *)
a[n_] := SeriesCoefficient[Tan[x + Pi/4], {x, 0, 2n}] // Denominator;
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 16 2023 *)

a(n) = denominator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023

a(n) = A046983(2*n).

(Zeta(2*n + 1, 1/4) - Zeta(2*n + 1, 3/4))/(Pi^(2*n + 1)) = A000364(n)*(2*n + 1)*2^(2*n)/(2*n + 1)!.

A361007 a(n) = numerator of (zeta(2n,1/4) + zeta(2n,3/4))/Pi^(2*n) where zeta is the Hurwitz zeta function.

Original entry on oeis.org

0, 2, 8, 64, 2176, 31744, 2830336, 178946048, 30460116992, 839461371904, 232711080902656, 39611984424992768, 955693069653508096, 1975371841521663868928, 1124025625663103358205952, 369906947004953565463576576, 278846808228005417477465964544

Offset: 0

Artur Jasinski, Mar 15 2023

The function (zeta(2*n,1/4) + zeta(2*n,3/4))/Pi^(2*n) is rational for every positive integer n.

			tan(2*x) = 2*x + (8/3)*x^3 + (64/15)*x^5 + (2176/315)*x^7 + (31744/2835)*x^9 + ...

Cf. A036279 (denominators).

Cf. A000364, A046982, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173982, A173983, A173984, A173987, A360945, A360966.

Table[(Zeta[2*n, 1/4] + Zeta[2*n, 3/4])/Pi^(2*n), {n, 0, 25}] //
  FunctionExpand // Numerator
Table[4^(2 k) (2^(2 k) - 1) (-1)^(k + 1) BernoulliB[2 k]/(2 (2 k)!), {k, 0, 25}] // Numerator

my(x='x+O('x^50), v = Vec(tan(2*x)/x)); apply(numerator, vector(#v\2, k, v[2*k-1])) \\ Michel Marcus, Apr 09 2023

a(n) = numerator( [x^(2*n-1)] tan(2*x) ).

a(n) = numerator( (-1)^(n + 1)*4^(2*n)*(2^(2*n) - 1)*B(2*n)/(2*(2*n)!) ) where B(2*n) are Bernoulli numbers.

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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A046983 Denominators of Taylor series for tan(x + Pi/4).

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A360945 a(n) = numerator of (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

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A360966 a(n) = denominator of (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

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Mathematica

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A361007 a(n) = numerator of (zeta(2*n,1/4) + zeta(2*n,3/4))/Pi^(2*n) where zeta is the Hurwitz zeta function.

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Author

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Comments

Examples

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Programs

Mathematica

PARI

Formula

A360945 a(n) = numerator of (Zeta(2n+1,1/4) - Zeta(2n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

A360966 a(n) = denominator of (Zeta(2n+1,1/4) - Zeta(2n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

A361007 a(n) = numerator of (zeta(2n,1/4) + zeta(2n,3/4))/Pi^(2*n) where zeta is the Hurwitz zeta function.