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

%I A328191 #33 Oct 07 2019 14:15:33
%S A328191 -1,1,1,5,19,-47,659,1837,7675,-154729,3578279,3984853,95259767,
%T A328191 -1364856587,85215100151,46900690817,4180679675171,-157452879103733,
%U A328191 6399968826052559,893237028559439,478887035449041839,-11642446645024341437,1123931378903214542099,15392026390952264029
%N A328191 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = numerator(f_n).
%C A328191 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.
%C A328191 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).
%C A328191 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.]
%C A328191 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".
%C A328191 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.
%H A328191 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 A328191 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 A328191 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 A328191 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 A328191 W. Lang, <a href="/A157162/a157162.txt">Recurrences for the general problem</a>, 2009.
%F A328191 a(2*n + 1) = -A170914(2*n + 1) for n >= 0.
%F A328191 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)).
%F A328191 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.
%e A328191 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, ...
%p A328191 # Calculates the fractions f_n (choose L much larger than M):
%p A328191 PPE := proc(L, M)
%p A328191 local t1, t0, g, t2, n, t3;
%p A328191 if L < 2.5*M then print("Choose larger value for L");
%p A328191 else
%p A328191 t1 := 1/(1 + sin(x));
%p A328191 t0 := series(t1, x, L);
%p A328191 f := []; t2 := t0;
%p A328191 for n to M do
%p A328191 t3 := coeff(t2, x, n);
%p A328191 t2 := series(t2/(1 + t3*x^n), x, L);
%p A328191 f := [op(f), t3];
%p A328191 end do;
%p A328191 end if;
%p A328191 [seq(f[n], n = 1 .. nops(f))];
%p A328191 end proc;
%p A328191 # Calculates the numerators of f_n:
%p A328191 h:=map(numer, PPE(100, 40)); # _Petros Hadjicostas_, Oct 07 2019 by modifying _N. J. A. Sloane_'s program from A170912 and A170913.
%t A328191 A[m_, n_] :=
%t A328191   A[m, n] =
%t A328191    Which[m == 1, 2*(-1)^n*I^(n + 2)*PolyLog[-(n + 1), -I]/n!,
%t A328191     m > n >= 1, 0, True,
%t A328191     A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
%t A328191 a[n_] := Numerator[A[n, n]];
%t A328191 a /@ Range[1, 55] (* _Petros Hadjicostas_, Oct 07 2019 using a program by _Jean-François Alcover_ and a formula from A099612 and A099617 *)
%Y A328191 Denominators are in A328186.
%Y A328191 Cf. A099612, A099617, A170914, A170915, A279107.
%K A328191 sign,frac
%O A328191 1,4
%A A328191 _Petros Hadjicostas_, Oct 07 2019