login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A328191 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = numerator(f_n). 3
-1, 1, 1, 5, 19, -47, 659, 1837, 7675, -154729, 3578279, 3984853, 95259767, -1364856587, 85215100151, 46900690817, 4180679675171, -157452879103733, 6399968826052559, 893237028559439, 478887035449041839, -11642446645024341437, 1123931378903214542099, 15392026390952264029 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..24.

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.

FORMULA

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.

EXAMPLE

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, ...

MAPLE

# 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.

MATHEMATICA

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-Fran├žois Alcover and a formula from A099612 and A099617 *)

CROSSREFS

Denominators are in A328186.

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

Sequence in context: A243895 A024191 A277801 * A100104 A015650 A200764

Adjacent sequences:  A328188 A328189 A328190 * A328192 A328193 A328194

KEYWORD

sign,frac

AUTHOR

Petros Hadjicostas, Oct 07 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 14 18:32 EDT 2020. Contains 336483 sequences. (Running on oeis4.)