A279107 -id:A279107 - cp's OEIS frontend

A099612 Numerators of the coefficients in the Taylor expansion of sec(x) + tan(x) around x=0.

1, 1, 1, 1, 5, 2, 61, 17, 277, 62, 50521, 1382, 540553, 21844, 199360981, 929569, 3878302429, 6404582, 2404879675441, 443861162, 14814847529501, 18888466084, 69348874393137901, 113927491862, 238685140977801337, 58870668456604, 4087072509293123892361

Offset: 0

N. J. A. Sloane, Nov 18 2004

			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

Seiichi Manyama, Table of n, a(n) for n = 0..487
L. Euler, On the sums of series of reciprocals, par. 13, arXiv:math/0506415 [math.HO], 2005-2008.
L. Euler, De summis serierum reciprocarum, E41, par. 13, Euler Archive.
Peter Luschny, The Euler-Bernoulli diamond and the lost Bernoulli numbers.

Cf. A000111, A099617, A279107.

R := n -> (cos(n*Pi/2)+sin(n*Pi/2))*(4^(n+1)-2^(n+1))*((Zeta(0,-n,3/4)-Zeta(0,-n,1/4))/(2^(-n)-2)-Zeta(-n))/GAMMA(n+1):
[1, seq(numer(R(n)), n=1..19)]; # Peter Luschny, Aug 25 2015
# From Peter Luschny, Aug 03 2017: (Start) By recurrence:
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: A099612 := n -> numer(S(n, n)/n!):
seq(A099612(n), n=0..26);
# or evaluating polynomials at -i:
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(numer(R(n)), n=0..26);
# or with the Euler polynomials:
ep := n -> `if`(n=0,1,2^n*(euler(n,1/2)-euler(n,1))*(-1)^iquo(n+1,2)):
a := n -> numer(ep(n)/n!): seq(a(n), n=0..26); # (End)

nn = 26; Numerator[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,0,26}] // Numerator (* Peter Luschny, Aug 03 2017 *)
Table[(1 + Mod[n,2])LerchPhi[(-1)^(n+1), n+1, 1/2]/Pi^(n+1), {n, 0, 26}] // Numerator (* Peter Luschny, Aug 03 2017 *)

Let R(x) = (-1)^floor(x/2)*(4^(x+1)-2^(x+1))*((HurwitzZeta(-x,3/4) - HurwitzZeta(-x,1/4)) /(2^(-x)-2)-Zeta(-x))/Gamma(x+1) then a(n) = numerator(R(n)) and A099617(n) = denominator(R(n)) for n>=1. - Peter Luschny, Aug 25 2015
Let F(x,t) = exp(-I*t*x)*(1+(exp(exp(I*t))-1)/(exp(2*exp(I*t))+1)) and r(x) = ((cos(x*Pi/2)+sin(x*Pi/2))/Pi)*Integral_{t=0..2*Pi} F(x,t) then a(n) = numerator(r(n)) and A099617(n) = denominator(r(n)) for n>=1. - Peter Luschny, Aug 25 2015
a(n)/A099617(n) = A000111(n)/n!. - Seiichi Manyama, Jan 27 2017
From Peter Luschny, Aug 03 2017: (Start)
a(n) = numerator(2*i^(n+1)*PolyLog(-n, -i)/n!) for n>0.
a(n) = numerator(2^n*|Euler(n,1/2) - Euler(n,1)|/n!) for n>0 where Euler(n,x) are the Euler polynomials. (End)
Conjecture: For n >= 0, (-1)^n * a(n+1) is the numerator 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)). Clark Kimberling in A279107 states my conjecture as a fact, but no proof or reference is given.] - Petros Hadjicostas, Oct 06 2019

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2004

			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

Seiichi Manyama, Table of n, a(n) for n = 0..451
L. Euler, On the sums of series of reciprocals, par. 13, arXiv:math/0506415 [math.HO], 2005-2008.
L. Euler, De summis serierum reciprocarum, E41, par. 13.
Peter Luschny, The Euler-Bernoulli diamond and the lost Bernoulli numbers.

Cf. A099612, A279107.

# 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)

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 *)

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)
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

A328186 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = denominator(f_n).

Original entry on oeis.org

1, 1, 6, 6, 120, 360, 5040, 2520, 72576, 1814400, 39916800, 59875200, 1245404160, 21794572800, 1307674368000, 81729648000, 71137485619200, 3201186852864000, 121645100408832000, 12164510040883200, 10218188434341888000, 281000181944401920000, 25852016738884976640000

Offset: 1

Petros Hadjicostas, Oct 06 2019

The recurrence about (A(m,n): m,n >= 1) in the Formula section follows from Theorem 3 in Gingold et al. (1988); see also Gingold and Knopfmacher (1995, p. 1222). A(m=1,n) equals the n-th coefficient of the Taylor expansion of 1/(1 + sin(x)). For that coefficient, we use a modification of a formula by Peter Luschny in the documentation of sequences A099612 and A099617.
Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n). We have A170914(n) = numerator(g_n) and A170915(n) = denominator(g_n).
Gingold and Knopfmacher (1995) and Alkauskas (2008, 2009) proved that f_n = -g_n for n odd, and Sum_{s|n} (-g_{n/s})^s/s = -Sum_{s|n} (-f_{n/s})^s/s. [We caution that different authors may use -g_n for g_n, or -f_n for f_n, or both.]
Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s. He connects inverse power product expansions to unital series associated to (infinite dimensional) Witt vectors and to the so-called "Somos transformation".
There are more formulas for f_n and g_n in the references listed below. In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.

			f_n = -1, 1, 1/6, 5/6, 19/120, -47/360, 659/5040, 1837/2520, 7675/72576, -154729/1814400, 3578279/39916800, 3984853/59875200, 95259767/1245404160, ...

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
W. Lang, Recurrences for the general problem, 2009.

Numerators are in A328191.

Cf. A099612, A099617, A170914, A170915, A279107.

# Calculates the fractions f_n (choose L much larger than M):
PPE := proc(L, M)
local t1, t0, g, t2, n, t3;
if L < 2.5*M then print("Choose larger value for L");
else
t1 := 1/(1 + sin(x));
t0 := series(t1, x, L);
f := []; t2 := t0;
for n to M do
t3 := coeff(t2, x, n);
t2 := series(t2/(1 + t3*x^n), x, L);
f := [op(f), t3];
end do;
end if;
[seq(f[n], n = 1 .. nops(f))];
end proc;
# Calculates the denominators of f_n:
h := map(denom, PPE(100, 40)); # Petros Hadjicostas, Oct 06 2019 by modifying N. J. A. Sloane's program from A170912 and A170913.

A[m_, n_] :=
  A[m, n] =
   Which[m == 1, 2*(-1)^n*I^(n + 2)*PolyLog[-(n + 1), -I]/n!,
    m > n >= 1, 0, True,
    A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
a[n_] := Denominator[A[n, n]];
a /@ Range[1, 55] (* Petros Hadjicostas, Oct 06 2019 using a program by Jean-François Alcover and a formula from A099612 and A099617 *)

a(2*n + 1) = A170915(2*n + 1) for n >= 0.
Define (A(m,n): n,m >= 1) by A(m=1, n) = 2 * (-1)^n * i^(n + 2) * PolyLog(-(n + 1), -i)/n! for n >= 1 (with i := sqrt(-1)), A(m,n) = 0 for m > n >= 1 (upper triangular), and A(m,n) = A(m-1,n) - A(m-1,m-1) * A(m,n-m+1) for n >= m >= 2. Then f_n = A(n,n) and thus a(n) = denominator(A(n,n)).
If we write 1 + sin x = Product_{n>=1} (1 + g_n * x^n) and we know (g_n: n >= 1), then f_n = -g_n + Sum_{s|n, s > 1} (1/s) * ((-f_{n/s})^s + (-g_{n/s})^s). This proves of course that f_n = -g_n for n odd.

A328191 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = numerator(f_n).

Original entry on oeis.org

-1, 1, 1, 5, 19, -47, 659, 1837, 7675, -154729, 3578279, 3984853, 95259767, -1364856587, 85215100151, 46900690817, 4180679675171, -157452879103733, 6399968826052559, 893237028559439, 478887035449041839, -11642446645024341437, 1123931378903214542099, 15392026390952264029

Offset: 1

Petros Hadjicostas, Oct 07 2019

The recurrence about (A(m,n): m,n >= 1) in the Formula section follows from Theorem 3 in Gingold et al. (1988); see also Gingold and Knopfmacher (1995, p. 1222). A(m=1,n) equals the n-th coefficient of the Taylor expansion of 1/(1 + sin(x)). For that coefficient, we use a modification of a formula by Peter Luschny in the documentation of sequences A099612 and A099617.
Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n). We have A170914(n) = numerator(g_n) and A170915(n) = denominator(g_n).
Gingold and Knopfmacher (1995) and Alkauskas (2008, 2009) proved that f_n = -g_n for n odd, and Sum_{s|n} (-g_{n/s})^s/s = -Sum_{s|n} (-f_{n/s})^s/s. [We caution that different authors may use -g_n for g_n, or -f_n for f_n, or both.]
Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s. He connects inverse power product expansions to unital series associated to (infinite dimensional) Witt vectors and to the so-called "Somos transformation".
There are more formulas for f_n and g_n in the references listed below. In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.

			f_n = -1, 1, 1/6, 5/6, 19/120, -47/360, 659/5040, 1837/2520, 7675/72576, -154729/1814400, 3578279/39916800, 3984853/59875200, 95259767/1245404160, ...

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
W. Lang, Recurrences for the general problem, 2009.

Denominators are in A328186.

Cf. A099612, A099617, A170914, A170915, A279107.

# Calculates the fractions f_n (choose L much larger than M):
PPE := proc(L, M)
local t1, t0, g, t2, n, t3;
if L < 2.5*M then print("Choose larger value for L");
else
t1 := 1/(1 + sin(x));
t0 := series(t1, x, L);
f := []; t2 := t0;
for n to M do
t3 := coeff(t2, x, n);
t2 := series(t2/(1 + t3*x^n), x, L);
f := [op(f), t3];
end do;
end if;
[seq(f[n], n = 1 .. nops(f))];
end proc;
# Calculates the numerators of f_n:
h:=map(numer, PPE(100, 40)); # Petros Hadjicostas, Oct 07 2019 by modifying N. J. A. Sloane's program from A170912 and A170913.

A[m_, n_] :=
  A[m, n] =
   Which[m == 1, 2*(-1)^n*I^(n + 2)*PolyLog[-(n + 1), -I]/n!,
    m > n >= 1, 0, True,
    A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
a[n_] := Numerator[A[n, n]];
a /@ Range[1, 55] (* Petros Hadjicostas, Oct 07 2019 using a program by Jean-François Alcover and a formula from A099612 and A099617 *)

a(2*n + 1) = -A170914(2*n + 1) for n >= 0.
Define (A(m,n): n,m >= 1) by A(m=1, n) = 2 * (-1)^n * i^(n + 2) * PolyLog(-(n + 1), -i)/n! for n >= 1 (with i := sqrt(-1)), A(m,n) = 0 for m > n >= 1 (upper triangular), and A(m,n) = A(m-1,n) - A(m-1,m-1) * A(m,n-m+1) for n >= m >= 2. Then f_n = A(n,n) and thus a(n) = numerator(A(n,n)).
If we write 1 + sin x = Product_{n>=1} (1 + g_n * x^n) and we know (g_n: n >= 1), then f_n = -g_n + Sum_{s|n, s > 1} (1/s) * ((-f_{n/s})^s + (-g_{n/s})^s). This proves of course that f_n = -g_n for n odd.

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A099612 Numerators of the coefficients in the Taylor expansion of sec(x) + tan(x) around x=0.

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

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A328186 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = denominator(f_n).

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A328191 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = numerator(f_n).

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