A050970 -id:A050970 - cp's OEIS frontend

A050971 4*Denominator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} ((4k+1)^(-n)).

1, 2, 8, 24, 384, 240, 46080, 40320, 2064384, 725760, 3715891200, 159667200, 392398110720, 12454041600, 1428329123020800, 20922789888000, 274239191619993600, 711374856192000, 1678343852714360832000

Offset: 1

Eric W. Weisstein

Reduced denominators of the Favard constants.

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

Theo Niessink, Table of n, a(n) for n = 1..200 (uploaded again by _Georg Fischer_, Feb 20 2019)
N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001-2003.
N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
Maths StackExchange, Can this equation be written in terms of x?, Apr 22 2021.
Eric Weisstein's World of Mathematics, Favard Constants.

Cf. A068205, A050970 (numerators).

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!):
A050971 := n -> denom(EZ(n-1)): seq(A050971(n), n=1..19); # Peter Luschny, Aug 02 2017

s[n_] := Sum[(4*k + 1)^(-n), {k, -Infinity, Infinity}]; a[n_] := 4*s[n]/Pi^n ; a[1] = 1; Table[a[n], {n, 1, 19}] // Denominator (* Jean-François Alcover, Nov 05 2012 *)
a[n_] := 4*Sum[((-1)^k/(2*k+1))^n, {k, 0, Infinity}] /. Pi -> 1 // Denominator; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 20 2014 *)
Table[4/(2 Pi)^n LerchPhi[(-1)^n, n, 1/2], {n, 21}] // Denominator (* Eric W. Weisstein, Aug 02 2017 *)
Table[4/Pi^n If[Mod[n, 2] == 0, DirichletLambda, DirichletBeta][n], {n, 21}] // Denominator (* Eric W. Weisstein, Aug 02 2017 *)

There is a simple formula in terms of Euler and Bernoulli numbers.

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

A089171 Numerators of series coefficients of 1/(1 + cosh(sqrt(x))).

Original entry on oeis.org

1, -1, 1, -17, 31, -691, 5461, -929569, 3202291, -221930581, 4722116521, -56963745931, 14717667114151, -2093660879252671, 86125672563201181, -129848163681107301953, 868320396104950823611, -209390615747646519456961, 14129659550745551130667441

Offset: 0

Wouter Meeussen, Dec 07 2003

Unsigned version is equal to A002425 up to n=11, but differs beyond that point.

Unsigned version: numerators of series coefficients of 1/(1 + cos(sqrt(x))); see Mathematica. - Clark Kimberling, Dec 06 2016

N. J. A. Sloane, Table of n, a(n) for n = 0..299

Cf. A002425, A050970, A002430, A046990.
Cf. A000182, A198631, A279009, A279010.
Cf. A276592, A279370.

with(numtheory): c := n->(2^(2*n)-1)*bernoulli(2*n)/(2*n)!; seq(numer(c(n)),n=1..20); # C. Ronaldo

Numerator[CoefficientList[Series[1/(1+Cosh[Sqrt[x]]), {x, 0, 24}], x]]
Numerator[CoefficientList[Series[1/(1+Cos[Sqrt[x]]), {x, 0, 30}], x]]
(* unsigned version, Clark Kimberling, Dec 06 2016 *)

a(n) = numerator(c(n+1)) where c(n)=(2^(2*n)-1)*B(2*n)/(2*n)!, B(k) denotes the k-th Bernoulli number. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004
Numerators of expansion of cosec(x)-cot(x) = 1/2*x+1/4*x^3/3!+1/2*x^5/5!+17/8*x^7/7!+31/2*x^9/9!+... - Ralf Stephan, Dec 21 2004 (Comment was applied to wrong entry, corrected by Alessandro Musesti (musesti(AT)gmail.com), Nov 02 2007)
E.g.f.: 1/sin(x)-cot(x). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: x/G(0); G(k) = 4*k+2-x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: (1+x/(x-2*Q(0)))/2; Q(k) = 8*k+2+x/(1+(2*k+1)*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: x/(x+Q(0)); Q(k) = x+(x^2)/((4*k+1)*(4*k+2)-(4*k+1)*(4*k+2)/(1+(4*k+3)*(4*k+4)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: T(0)/2, where T(k) = 1 - x^2/(x^2 - (4*k+2)*(4*k+6)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2013
Aerated, these are the numerators of the Taylor series coefficients of 2 * tanh(x/2) (cf. A000182 and A198631). - Tom Copeland, Oct 19 2016

A046990 Numerators of Taylor series for log(1/cos(x)). Also from log(cos(x)).

Original entry on oeis.org

0, 1, 1, 1, 17, 31, 691, 10922, 929569, 3202291, 221930581, 9444233042, 56963745931, 29435334228302, 2093660879252671, 344502690252804724, 129848163681107301953, 868320396104950823611, 209390615747646519456961, 28259319101491102261334882

Offset: 0

N. J. A. Sloane

			log(1/cos(x)) = 1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...
log(cos(x)) = -(1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:3 at page 301.

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

Cf. A046991, A002430, A050970.

q:= proc(n) add((-1)^k*combinat[eulerian1](n-1,k), k=0..n-1) end: A046990:= n -> numer((-1)^(n-1)*q(2*n)/(2*n)!):
seq(A046990(n),n=0..19);  # Peter Luschny, Nov 16 2012

Join[{0},Numerator[Select[CoefficientList[Series[Log[1/Cos[x]],{x,0,40}], x],#!=0&]]] (* Harvey P. Dale, Jul 27 2011 *)
a[n_] := Numerator[((-4)^n-(-16)^n)*BernoulliB[2*n]/2/n/(2*n)!]; a[0] = 0; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 11 2014, after Charles R Greathouse IV *)

a(n)=numerator(((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!) \\ Charles R Greathouse IV, Nov 06 2013

{a(n) = if( n<1, 0, my(m = 2*n); numerator( polcoeff( -log(cos(x + x * O(x^m))), m)))}; /* Michael Somos, Jun 03 2019 */

# uses[eulerian1 from A173018]
def A046990(n):
    def q(n):
        return add((-1)^k*eulerian1(n-1, k) for k in (0..n-1))
    return ((-1)^(n-1)*q(2*n)/factorial(2*n)).numer()
[A046990(n) for n in (0..19)]  # Peter Luschny, Nov 16 2012

Let q(n) = Sum_{k=0..n-1} (-1)^k*A201637(n-1,k) then a(n) = numerator((-1)^(n-1)*q(2*n)/(2*n)!). - Peter Luschny, Nov 16 2012

A245198 Decimal expansion of the Landau-Kolmogorov constant C(3,1) for derivatives in the case L_infinity(-infinity, infinity).

Original entry on oeis.org

1, 0, 4, 0, 0, 4, 1, 9, 1, 1, 5, 2, 5, 9, 5, 2, 0, 5, 7, 2, 6, 5, 0, 2, 8, 4, 1, 2, 1, 7, 8, 9, 4, 2, 6, 9, 3, 1, 6, 8, 9, 0, 2, 6, 7, 0, 1, 8, 6, 6, 3, 1, 0, 5, 4, 8, 4, 8, 7, 9, 5, 5, 4, 0, 1, 0, 0, 0, 5, 3, 1, 5, 5, 6, 9, 8, 6, 3, 4, 3, 8, 6, 8, 0, 3, 0, 2, 8, 3, 1, 8, 3, 9, 5, 3, 7, 8, 7, 4, 3, 3, 6, 4, 3

Offset: 1

Jean-François Alcover, Jul 17 2014

The corresponding Landau-Kolmogorov inequality for the first and third derivative is ||f'|| <= C(3,1) ||f||^(2/3) ||f'''||^(1/3) [see S. Finch ref. for C(n,k) and the general derivative inequalities], where the real-valued function f is defined on (-infinity, infinity), the involved norm being the supremum norm, defined by ||f|| = sup |f(x)|.
Hadamard proved that if f is twice differentiable and both f and f'' are bounded, then ||f'|| <= sqrt(2) ||f||^(1/2) ||f''||^(1/2), and the constant C(2,1) = sqrt(2) is the best possible.
Kolmogorov determined best constants C(n,k), 1 <= k <= n, for the inequality between derivatives in terms of Favard constants (A050970/A050971). These formulas giving C(n,k) include special cases discovered by G. E. Shilov for small values of n and k.
[All comments made after Steven R. Finch].

			1.0400419115259520572650284121789426931689026701866310548487955401...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[3, 1], 10, 104] // First
(* or, directly: *) RealDigits[3^(2/3)/2, 10, 104] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(3,1) = 3^(2/3)/2 = (9/8)^(1/3).

A046982 Numerators of Taylor series for tan(x + Pi/4).

Original entry on oeis.org

1, 2, 2, 8, 10, 64, 244, 2176, 554, 31744, 202084, 2830336, 2162212, 178946048, 1594887848, 30460116992, 7756604858, 839461371904, 9619518701764, 232711080902656, 59259390118004, 39611984424992768, 554790995145103208, 955693069653508096

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. A046983.

a(n) = 2^k * A050970(n), for some k>=0 (conjectured).

nmax = 23; 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]; Numerator[ 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]//Numerator (* Harvey P. Dale, May 21 2023 *)

A068205 Denominator of S(n)/Pi^n, where S(n) = Sum((4k+1)^(-n),k,-inf,+inf).

Original entry on oeis.org

4, 8, 32, 96, 1536, 960, 184320, 161280, 8257536, 2903040, 14863564800, 638668800, 1569592442880, 49816166400, 5713316492083200, 83691159552000, 1096956766479974400, 2845499424768000, 6713375410857443328000

Offset: 1

N. J. A. Sloane, Mar 24 2002

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

N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001-2003.
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

Numerators: A050970.

A068205 := 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 ;
    denom(%) ;
end proc:
seq(A068205(n),n=1..20) ; # R. J. Mathar, Jun 26 2024

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] // Denominator, {n, 1, 19}] (* Jean-François Alcover, May 13 2013 *)
a[n_] := Sum[((-1)^k/(2*k+1))^n, {k, 0, Infinity}] /. Pi -> 1 // Denominator; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 20 2014 *)

There is a simple formula in terms of Euler and Bernoulli numbers.

A246006 a(2n) = numerator of |Bernoulli(2n)|, a(2n+1) = Euler(2n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 61, 1, 1385, 5, 50521, 691, 2702765, 7, 199360981, 3617, 19391512145, 43867, 2404879675441, 174611, 370371188237525, 854513, 69348874393137901, 236364091, 15514534163557086905, 8553103, 4087072509293123892361, 23749461029, 1252259641403629865468285

Offset: 0

Eric Chen, Nov 13 2014

nonn

Primes p which divide at least one a(n) for n<=p-2 are called weakly-irregular primes. For example, 19|a(11), 31|a(23), 37|a(32), 43|a(13), 47|a(15), 59|a(44), 61|a(7), ... - Eric Chen, Nov 26 2014
The weakly-irregular primes below 500 are 19, 31, 37, 43, 47, 59, 61, 67, 71, 79, 101, 103, 131, 137, 139, 149, 157, 193, 223, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 347, 349, 353, 359, 373, 379, 389, 401, 409, 419, 421, 433, 461, 463, 467, 491. - Eric Chen, Nov 26 2014
A prime can divide more than one a(n) for n<=p-2; for example, 67 divides both a(27) and a(58); additional examples are 101, 149, 157, 241, 263, 307, 311, ... . - Eric Chen, Nov 26 2014
Smallest values of k such that the n-th weakly-irregular prime divides a(k) are 11, 23, 32, 13, 15, 44, 7, 27, 29, 19, 63, 24, 22, 43, 129, 130, 62, 75, ... . - Eric Chen, Nov 26 2014
Smallest prime factors (>= n+2) of a(n) are 1, 1, 1, 1, 1, 1, 1, 61, 1, 277, 1, 19, 691, 43, 1, 47, 3617, 228135437, 43867, 79, 283, 41737, 131, 31, 103, 2137, 657931, 67, 9349, 71, ... . - Eric Chen, Nov 26 2014
The irregular pairs are (61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ... . - Eric Chen, Nov 26 2014

			Euler(10) = 50521, so a(11) = 50521.
Bernoulli(12) = 691/2730, so a(12) = 691.

Eric Chen, Table of n, a(n) for n = 0..199
Peter Luschny, Irregular primes
Wikipedia, Irregular prime

Cf. A027641, A122045, A000367, A028296, A000111, A000364, A000182.

Cf. A050970, A050971, A068205.

a246006[n_] := If[EvenQ[n], Abs[Numerator[BernoulliB[n]]], Abs[EulerE[n-1]]]; Table[a246006[n], {n, 0, 99}]

from sympy import euler, bernoulli
def A246006(n): return abs(euler(n-1)) if n&1 else abs(bernoulli(n)).p # Chai Wah Wu, Apr 15 2023

A245294 Decimal expansion of the square root of 6/5.

Original entry on oeis.org

1, 0, 9, 5, 4, 4, 5, 1, 1, 5, 0, 1, 0, 3, 3, 2, 2, 2, 6, 9, 1, 3, 9, 3, 9, 5, 6, 5, 6, 0, 1, 6, 0, 4, 2, 6, 7, 9, 0, 5, 4, 8, 9, 3, 8, 9, 9, 9, 5, 9, 6, 6, 5, 0, 8, 4, 5, 3, 7, 8, 8, 8, 9, 9, 4, 6, 4, 9, 8, 6, 5, 5, 4, 2, 4, 5, 4, 4, 5, 4, 6, 7, 6, 0, 1, 7, 1, 6, 8, 7, 2, 3, 2, 7, 7, 4, 1, 2, 5, 1, 5, 2, 9, 4, 5

Offset: 1

Jean-François Alcover, Jul 17 2014

Decimal expansion of the Landau-Kolmogorov constant C(4,2) for derivatives in the case L_infinity(infinity, infinity).
See A245198.
Apart from the first digit the same as A176057. - R. J. Mathar, Jul 21 2014

			1.095445115010332226913939565601604267905489389995966508453788899464986554...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Eric Weisstein's MathWorld, Landau-Kolmogorov Constants.
Eric Weisstein's MathWorld, Favard Constants.

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[4, 2], 10, 105] // First
RealDigits[Sqrt[6/5], 10, 100][[1]] (* Amiram Eldar, Jul 19 2022 *)

sqrt(6/5) \\ Charles R Greathouse IV, Aug 26 2017

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).
C(4,2) = sqrt(6/5).
Equals Sum_{k>=0} binomial(2*k,k)/24^k. - Amiram Eldar, Jul 19 2022

A245293 Decimal expansion of the Landau-Kolmogorov constant C(4,1) for derivatives in the case L_infinity(infinity, infinity).

Original entry on oeis.org

1, 0, 8, 0, 9, 6, 0, 1, 2, 3, 8, 4, 5, 6, 2, 7, 5, 1, 5, 1, 8, 8, 0, 8, 0, 1, 5, 0, 6, 3, 6, 5, 4, 5, 6, 4, 9, 2, 3, 7, 5, 7, 7, 0, 7, 4, 7, 2, 5, 5, 2, 3, 4, 3, 8, 0, 1, 3, 5, 6, 6, 4, 4, 2, 5, 9, 2, 7, 5, 9, 9, 0, 9, 7, 9, 0, 6, 6, 8, 5, 7, 2, 5, 0, 6, 8, 4, 8, 1, 8, 1, 1, 2, 7, 0, 7, 0, 7, 6, 1, 6, 1, 7, 7, 9

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.0809601238456275151880801506365456492375770747255234380135664425927599...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[4, 1], 10, 105] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(4,1) = 4*(2/3)^(1/4)/5^(3/4) = (512/375)^(1/4).

A245295 Decimal expansion of the Landau-Kolmogorov constant C(4,3) for derivatives in the case L_infinity(infinity, infinity).

Original entry on oeis.org

1, 4, 8, 0, 1, 6, 5, 6, 0, 8, 9, 8, 4, 5, 7, 0, 5, 0, 1, 1, 3, 3, 5, 7, 9, 9, 3, 2, 3, 2, 7, 6, 7, 3, 6, 3, 8, 5, 9, 8, 1, 2, 3, 5, 8, 2, 6, 1, 2, 3, 7, 6, 2, 3, 6, 6, 4, 9, 7, 2, 4, 8, 1, 1, 8, 3, 1, 4, 9, 3, 3, 7, 3, 1, 5, 9, 9, 2, 3, 0, 5, 2, 4, 0, 8, 8, 8, 3, 9, 0, 3, 8, 0, 3, 7, 6, 7, 9, 7, 3, 4, 5, 1, 4, 9

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.480165608984570501133579932327673638598123582612376236649724811831493373...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[4, 3], 10, 105] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(4,3) = (3/5)^(1/4)*2^(3/4) = (24/5)^(1/4).

cp's OEIS Frontend

A050971 4*Denominator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} ((4k+1)^(-n)).

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A089171 Numerators of series coefficients of 1/(1 + cosh(sqrt(x))).

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A046990 Numerators of Taylor series for log(1/cos(x)). Also from log(cos(x)).

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A245198 Decimal expansion of the Landau-Kolmogorov constant C(3,1) for derivatives in the case L_infinity(-infinity, infinity).

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A046982 Numerators of Taylor series for tan(x + Pi/4).

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A068205 Denominator of S(n)/Pi^n, where S(n) = Sum((4k+1)^(-n),k,-inf,+inf).

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A246006 a(2n) = numerator of |Bernoulli(2n)|, a(2n+1) = Euler(2n).

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