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A099617 Denominators of the coefficients in the Taylor expansion of sec(x) + tan(x) around x=0.

%I A099617 #38 Jun 22 2025 04:14:33
%S A099617 1,1,2,3,24,15,720,315,8064,2835,3628800,155925,95800320,6081075,
%T A099617 87178291200,638512875,4184557977600,10854718875,6402373705728000,
%U A099617 1856156927625,97316080327065600,194896477400625,1124000727777607680000,2900518163668125,9545360026665222144000
%N A099617 Denominators of the coefficients in the Taylor expansion of sec(x) + tan(x) around x=0.
%H A099617 Seiichi Manyama, <a href="/A099617/b099617.txt">Table of n, a(n) for n = 0..451</a>
%H A099617 L. Euler, <a href="https://arxiv.org/abs/math/0506415">On the sums of series of reciprocals</a>, par. 13, arXiv:math/0506415 [math.HO], 2005-2008.
%H A099617 L. Euler, <a href="http://eulerarchive.maa.org">De summis serierum reciprocarum</a>, E41, par. 13.
%H A099617 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostBernoulliNumbers#Formulas_for_Euler.27s_Zeta_numbers">The Euler-Bernoulli diamond and the lost Bernoulli numbers</a>.
%F A099617 A099612(n)/a(n) = A000111(n)/n!. - _Seiichi Manyama_, Jan 27 2017
%F A099617 From _Peter Luschny_, Aug 03 2017: (Start)
%F A099617 a(n) = denominator(2*i^(n+1)*PolyLog(-n, -i)/n!) for n > 0.
%F A099617 a(n) = denominator(2^n*|Euler(n,1/2) - Euler(n,1)|/n!) for n > 0 where Euler(n,x) are the Euler polynomials. (End)
%F A099617 Conjecture: For n >= 0, (-1)^n * a(n+1)/(n+1) is the denominator of the n-th term of the Taylor expansion of 1/(1 + sin(x)) around x = 0. [This is based on the fact that (sec(x) + tan(x))' = 1/(1 + sin(-x)). Cf. also the comments in A099612 and A279107.] - _Petros Hadjicostas_, Oct 06 2019
%e A099617 1 + x + (1/2)*x^2 + (1/3)*x^3 + (5/24)*x^4 + (2/15)*x^5 + (61/720)*x^6 + (17/315)*x^7 + ...
%e A099617 1, 1, 1/2, 1/3, 5/24, 2/15, 61/720, 17/315, 277/8064, 62/2835, 50521/3628800, 1382/155925, 540553/95800320, ... = A099612/A099617
%p A099617 # From _Peter Luschny_, Aug 03 2017: (Start)
%p A099617 S := proc(n, k) option remember; if k = 0 then `if`(n = 0, 1, 0) else
%p A099617 S(n, k - 1) + S(n - 1, n - k) fi end: A099617 := n -> denom(S(n, n)/n!):
%p A099617 seq(A099617(n), n=0..24); # version 1
%p A099617 P := proc(n,x) local k,j; add(add((-1)^j*2^(-k)*binomial(k,j)
%p A099617 *(k-2*j)^n* x^(n-k), j=0..k), k=0..n) end: R := n -> `if`(n = 0, 1, P(n-1, -I)/ n!): seq(denom(R(n)), n=0..24); # version 2
%p A099617 ep := n -> `if`(n=0,1,2^n*abs(euler(n,1/2) - euler(n,1))):
%p A099617 a := n -> denom(ep(n)/n!): seq(a(n), n=0..24); # version 3 (End)
%t A099617 nn = 24; Denominator[CoefficientList[Series[Sec[x] + Tan[x], {x, 0, nn}], x]] (* _T. D. Noe_, Jul 24 2013 *)
%t A099617 Table[If[n==0,1,2 I ^(n+1) PolyLog[-n, -I] / n!], {n,0,24}] // Denominator (* _Peter Luschny_, Aug 03 2017 *)
%t A099617 Table[2 (1 + Mod[n, 2]) LerchPhi[(-1)^(n+1), n+1, 1/2] / Pi^(n+1), {n, 0, 24}] // Denominator (* _Peter Luschny_, Aug 03 2017 *)
%Y A099617 Cf. A099612, A279107.
%K A099617 nonn,frac
%O A099617 0,3
%A A099617 _N. J. A. Sloane_, Nov 19 2004