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A099617 Denominators of coefficients in expansion of sec(x) + tan(x). 4
1, 1, 2, 3, 24, 15, 720, 315, 8064, 2835, 3628800, 155925, 95800320, 6081075, 87178291200, 638512875, 4184557977600, 10854718875, 6402373705728000, 1856156927625, 97316080327065600, 194896477400625, 1124000727777607680000, 2900518163668125, 9545360026665222144000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..451

L. Euler, On the sums of series of reciprocals, par. 13.

L. Euler, De summis serierum reciprocarum, E41, par. 13.

Peter Luschny, The Euler-Bernoulli diamond and the lost Bernoulli numbers.

FORMULA

A099612(n)/a(n) = A000111(n)/n!. - Seiichi Manyama, Jan 27 2017

From Peter Luschny, Aug 03 2017: (Start)

a(n) = denominator(2*i^(n+1)*PolyLog(-n, -i)/n!) for n>0.

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)

EXAMPLE

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 + ...

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

MAPLE

# From Peter Luschny, Aug 03 2017: (Start)

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: A099617 := n -> denom(S(n, n)/n!):

seq(A099617(n), n=0..24); # version 1

P := proc(n, x) local k, j; add(add((-1)^j*2^(-k)*binomial(k, j)

*(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

ep := n -> `if`(n=0, 1, 2^n*abs(euler(n, 1/2) - euler(n, 1))):

a := n -> denom(ep(n)/n!): seq(a(n), n=0..24); # version 3 (End)

MATHEMATICA

nn = 24; Denominator[CoefficientList[Series[Sec[x] + Tan[x], {x, 0, nn}], x]] (* T. D. Noe, Jul 24 2013 *)

Table[If[n==0, 1, 2 I ^(n+1) PolyLog[-n, -I] / n!], {n, 0, 24}] // Denominator (* Peter Luschny, Aug 03 2017 *)

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 *)

CROSSREFS

Cf. A099612.

Sequence in context: A119619 A170909 A160606 * A092043 A055067 A037319

Adjacent sequences:  A099614 A099615 A099616 * A099618 A099619 A099620

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Nov 19 2004

STATUS

approved

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Last modified September 20 03:24 EDT 2017. Contains 292251 sequences.