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%I #56 Oct 09 2019 02:52:16
%S 1,0,-1,1,-19,19,-659,37,-7675,40043,-3578279,1123009,-95259767,
%T 7091713,-85215100151,832857559,-4180679675171,63804880881241,
%U -6399968826052559,5697831990097981,-478887035449041839,252737248941887573,-1123931378903214542099,35703551772944759
%N Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n); a(n) = numerator(g_n).
%C From _Petros Hadjicostas_, Oct 06 2019: (Start)
%C 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).
%C 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).
%C _Wolfdieter Lang_ (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s.
%C In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.
%C (End)
%H Giedrius Alkauskas, <a href="http://arxiv.org/abs/0801.0805">One curious proof of Fermat's little theorem</a>, arXiv:0801.0805 [math.NT], 2008.
%H Giedrius Alkauskas, <a href="https://www.jstor.org/stable/40391097">A curious proof of Fermat's little theorem</a>, Amer. Math. Monthly 116(4) (2009), 362-364.
%H H. Gingold, H. W. Gould, and Michael E. Mays, <a href="https://www.researchgate.net/publication/268023169_Power_product_expansions">Power Product Expansions</a>, Utilitas Mathematica 34 (1988), 143-161.
%H H. Gingold and A. Knopfmacher, <a href="http://dx.doi.org/10.4153/CJM-1995-062-9">Analytic properties of power product expansions</a>, Canad. J. Math. 47 (1995), 1219-1239.
%H W. Lang, <a href="/A157162/a157162.txt">Recurrences for the general problem</a>, 2009.
%F From _Petros Hadjicostas_, Oct 07 2019: (Start)
%F a(2*n+1) = -A328191(2*n+1) for n >= 0.
%F 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)
%e g_n = 1, 0, -1/6, 1/6, -19/120, 19/120, -659/5040, 37/280, -7675/72576, ...
%p # Calculates the fractions g_n (choose L much larger than M):
%p PPE_sin := proc(L, M)
%p local t1, t0, g, t2, n, t3;
%p if L < 2.5*M then print("Choose larger value for L");
%p else
%p t1 := 1 + sin(x);
%p t0 := series(t1, x, L);
%p g := []; t2 := t0;
%p for n to M do
%p t3 := coeff(t2, x, n);
%p t2 := series(t2/(1 + t3*x^n), x, L);
%p g := [op(g), t3];
%p end do;
%p end if;
%p [seq(g[n], n = 1 .. nops(g))];
%p end proc;
%p # Calculates the numerators of g_n:
%p h1 := map(numer, PPE_sin(100, 40)); # _Petros Hadjicostas_, Oct 06 2019 by modifying _N. J. A. Sloane_'s program from A170912 and A170913.
%t A[m_, n_] :=
%t A[m, n] =
%t Which[m == 1, (1-(-1)^n)*(-1)^Floor[(n-1)/2]/(2*n!), m > n >= 1, 0, True,
%t A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
%t a[n_] := Numerator[A[n, n]];
%t a /@ Range[1, 55] (* _Petros Hadjicostas_, Oct 06 2019, courtesy of _Jean-François Alcover_ *)
%Y Cf. Denominators are in A170915.
%Y Cf. A170910, A170911, A170912, A170913, A328186, A328191.
%K sign,frac
%O 1,5
%A _N. J. A. Sloane_, Jan 30 2010
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