A276592 -id:A276592 - cp's OEIS frontend

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

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

A276593 Denominator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).

Original entry on oeis.org

8, 96, 960, 161280, 2903040, 638668800, 49816166400, 83691159552000, 2845499424768000, 1946321606541312000, 408727537373675520000, 48662619743783485440000, 124089680346647887872000000, 174221911206693634572288000000, 70734095949917615636348928000000

Offset: 1

Martin Renner, Sep 07 2016

A276592(n)/a(n) * Pi^(2*n) = Sum_{k>=1} 1/(2*k-1)^(2*n) > 1. So Pi^(2*n) > a(n)/A276592(n). - Seiichi Manyama, Sep 03 2018

			From _Seiichi Manyama_, Sep 03 2018: (Start)
n |    Pi^(2*n)   |   a(n)/A276592(n)
--+---------------+------------------------------------
1 |        9.8... |           8
2 |       97.4... |          96
3 |      961.3... |         960
4 |     9488.5... |      161280/17     =     9487.0...
5 |    93648.0... |     2903040/31     =    93646.4...
6 |   924269.1... |   638668800/691    =   924267.4...
7 |  9122171.1... | 49816166400/5461   =  9122169.2... (End)

Seiichi Manyama, Table of n, a(n) for n = 1..225

Cf. A002432, A046988, A276592, A276594, A276595.

seq(denom(sum(1/(2*k-1)^(2*n),k=1..infinity)/Pi^(2*n)),n=1..22);

a[n_]:=Denominator[(1-2^(-2 n)) Zeta[2 n]] (* Steven Foster Clark, Mar 10 2023 *)
a[n_]:=Denominator[1/2 SeriesCoefficient[1/(E^x+1),{x,0,2 n-1}]] (* Steven Foster Clark, Mar 10 2023 *)
a[n_]:=Denominator[1/2 Residue[Zeta[s] Gamma[s] (1-2^(1-s)) x^(-s),{s,1-2 n}]] (* Steven Foster Clark, Mar 11 2023 *)

A276592(n)/a(n) + A276594(n)/A276595(n) = A046988(n)/A002432(n).

A276592(n)/a(n) = (-1)^(n+1) * B_{2*n} * (2^(2*n) - 1) / (2 * (2*n)!), where B_n is the Bernoulli number. - Seiichi Manyama, Sep 03 2018

A276595 Denominator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2k)^(2n).

Original entry on oeis.org

24, 1440, 60480, 2419200, 95800320, 2615348736000, 149448499200, 21341245685760000, 10218188434341888000, 1605715325396582400000, 28202200078783610880000, 3387648273463487338905600000, 372269041039943663616000000, 75786531374911731038945280000000

Offset: 1

Martin Renner, Sep 07 2016

Denominator of Bernoulli(2*n)/(2*(2*n)!). - Robert Israel, Sep 18 2016

Robert Israel, Table of n, a(n) for n = 1..223

Cf. A002432, A046988, A276592, A276593, A276594.

seq(denom(sum(1/(2*k)^(2*n),k=1..infinity)/Pi^(2*n)),n=1..24);
seq(denom(bernoulli(2*n)/2/(2*n)!),n=1..24); # Robert Israel, Sep 18 2016

Table[Denominator[Zeta[2*n]/(2*Pi)^(2*n)], {n, 1, 30}] (* Terry D. Grant, Jun 19 2018 *)

a(n) = denominator(bernfrac(2*n)/(2*(2*n)!)); \\ Michel Marcus, Jul 05 2018

A276592(n)/A276593(n) + A276594(n)/a(n) = A046988(n)/A002432(n).

Zeta(2n) = (-1)^(n-1)*(A276594(n)/a(n))*((2*Pi)^(2n)), according to Euler. - Terry D. Grant, Jun 19 2018

A276594 Numerator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2k)^(2n).

Original entry on oeis.org

1, 1, 1, 1, 1, 691, 1, 3617, 43867, 174611, 77683, 236364091, 657931, 3392780147, 1723168255201, 7709321041217, 151628697551, 26315271553053477373, 154210205991661, 261082718496449122051, 1520097643918070802691, 2530297234481911294093

Offset: 1

Martin Renner, Sep 07 2016

Cf. A002432, A046988, A276592, A276593, A276595.

seq(numer(sum(1/(2*k)^(2*n),k=1..infinity)/Pi^(2*n)),n=1..24);

A276592(n)/A276593(n) + a(n)/A276595(n) = A046988(n)/A002432(n).

A279370 Numerators of coefficients in expansion of (cos(sqrt(x)))/(1 + cos(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

Clark Kimberling, Dec 12 2016

Differs from A089171 in signs; see Formula.

			(1/2) - (1/8)x - (1/48)x^2 - (17/5760)x^3 + ... ; 1/2, - 1/8, - 48/2, - 17/5760, ... = A279370/A279109.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A089171, A279109, A279239, A276592.

z = 26; t = CoefficientList[Series[Cos[Sqrt[x]]/(1 + Cos[Sqrt[x]]), {x, 0, z}], x];
Numerator[t]   (* A279370 *)
Denominator[t] (* A279109 *)

For odd n and for n = 0, we have a(n) = A089171(n); for positive even n, however, a(n) = -A089171(n)

cp's OEIS Frontend

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

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A276593 Denominator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2*k-1)^(2*n).

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A276595 Denominator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2*k)^(2*n).

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A276594 Numerator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2*k)^(2*n).

Views

Author

Keywords

Crossrefs

Programs

Maple

Formula

A279370 Numerators of coefficients in expansion of (cos(sqrt(x)))/(1 + cos(sqrt(x))).

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Examples

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Programs

Mathematica

Formula

A276593 Denominator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).

A276595 Denominator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2k)^(2n).

A276594 Numerator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2k)^(2n).