A279370 -id:A279370 - 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

A276592 Numerator 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

1, 1, 1, 17, 31, 691, 5461, 929569, 3202291, 221930581, 4722116521, 56963745931, 14717667114151, 2093660879252671, 86125672563201181, 129848163681107301953, 868320396104950823611, 209390615747646519456961, 14129659550745551130667441, 16103843159579478297227731

Offset: 1

Martin Renner, Sep 07 2016

Apart from signs, same as A089171 and A279370. - Peter Bala, Feb 07 2019

Seiichi Manyama, Table of n, a(n) for n = 1..276
Siddharth Dwivedi, Vivek Kumar Singh, and Abhishek Roy, Semiclassical limit of topological Rényi entropy in 3d Chern-Simons theory, arXiv:2007.07033 [hep-th], 2020. See also J. of High Energy Physics (2020) Vol. 2020, Issue 12, Article 132.

Cf. A002432, A046988, A276593, A276594, A276595, A089171, A279370.

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

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

a(n)/A276593(n) + A276594(n)/A276595(n) = A046988(n)/A002432(n).

a(n)/A276593(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

cp's OEIS Frontend

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

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A276592 Numerator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A276592 Numerator 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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A276592 Numerator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).