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%I #33 Oct 07 2019 14:15:33
%S -1,1,1,5,19,-47,659,1837,7675,-154729,3578279,3984853,95259767,
%T -1364856587,85215100151,46900690817,4180679675171,-157452879103733,
%U 6399968826052559,893237028559439,478887035449041839,-11642446645024341437,1123931378903214542099,15392026390952264029
%N Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = numerator(f_n).
%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/(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 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 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 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 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 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 a(2*n + 1) = -A170914(2*n + 1) for n >= 0.
%F 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 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 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 # Calculates the fractions f_n (choose L much larger than M):
%p PPE := 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/(1 + sin(x));
%p t0 := series(t1, x, L);
%p f := []; 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 f := [op(f), t3];
%p end do;
%p end if;
%p [seq(f[n], n = 1 .. nops(f))];
%p end proc;
%p # Calculates the numerators of f_n:
%p h:=map(numer, PPE(100, 40)); # _Petros Hadjicostas_, Oct 07 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, 2*(-1)^n*I^(n + 2)*PolyLog[-(n + 1), -I]/n!,
%t 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 07 2019 using a program by _Jean-François Alcover_ and a formula from A099612 and A099617 *)
%Y Denominators are in A328186.
%Y Cf. A099612, A099617, A170914, A170915, A279107.
%K sign,frac
%O 1,4
%A _Petros Hadjicostas_, Oct 07 2019
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