A170914 Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n); a(n) = numerator(g_n).

1, 0, -1, 1, -19, 19, -659, 37, -7675, 40043, -3578279, 1123009, -95259767, 7091713, -85215100151, 832857559, -4180679675171, 63804880881241, -6399968826052559, 5697831990097981, -478887035449041839, 252737248941887573, -1123931378903214542099, 35703551772944759

Offset: 1

Author

N. J. A. Sloane, Jan 30 2010

Keywords

• sign
• frac

Comments

From Petros Hadjicostas, Oct 06 2019: (Start)

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 + sin(x).

If 1 + sin(x) = 1/Product_{n>=1} (1 + f_n * x^n) (inverse power product expansion), then 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.] We have A328191(n) = numerator(f_n) and A328186(n) = denominator(f_n).

Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s.

In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.

(End)

Examples

			g_n = 1, 0, -1/6, 1/6, -19/120, 19/120, -659/5040, 37/280, -7675/72576, ...
		

Links

• 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.

Crossrefs

Cf. Denominators are in A170915.

Cf. A170910, A170911, A170912, A170913, A328186, A328191.

Programs

Maple

# Calculates the fractions g_n (choose L much larger than M):
PPE_sin := 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 + sin(x);
t0 := series(t1, x, L);
g := []; t2 := t0;
for n to M do
t3 := coeff(t2, x, n);
t2 := series(t2/(1 + t3*x^n), x, L);
g := [op(g), t3];
end do;
end if;
[seq(g[n], n = 1 .. nops(g))];
end proc;
# Calculates the numerators of g_n:
h1 := map(numer, PPE_sin(100, 40)); # Petros Hadjicostas, Oct 06 2019 by modifying N. J. A. Sloane's program from A170912 and A170913.

Mathematica

A[m_, n_] :=
  A[m, n] =
   Which[m == 1, (1-(-1)^n)*(-1)^Floor[(n-1)/2]/(2*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 06 2019, courtesy of Jean-François Alcover *)

Formula

From Petros Hadjicostas, Oct 07 2019: (Start)

a(2*n+1) = -A328191(2*n+1) for n >= 0.

Define (A(m,n): n,m >= 1) by A(m=1,2*n+1) = (-1)^n/(2*n+1)! for n >= 0, A(m=1,2*n) = 0 for n >= 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 g_n = A(n,n). (End)