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%I A328186 #45 Oct 07 2019 14:15:09
%S A328186 1,1,6,6,120,360,5040,2520,72576,1814400,39916800,59875200,1245404160,
%T A328186 21794572800,1307674368000,81729648000,71137485619200,
%U A328186 3201186852864000,121645100408832000,12164510040883200,10218188434341888000,281000181944401920000,25852016738884976640000
%N A328186 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = denominator(f_n).
%C A328186 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 A328186 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 A328186 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 A328186 _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 A328186 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 A328186 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 A328186 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 A328186 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 A328186 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 A328186 W. Lang, <a href="/A157162/a157162.txt">Recurrences for the general problem</a>, 2009.
%F A328186 a(2*n + 1) = A170915(2*n + 1) for n >= 0.
%F A328186 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) = denominator(A(n,n)).
%F A328186 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 A328186 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 A328186 # Calculates the fractions f_n (choose L much larger than M):
%p A328186 PPE := proc(L, M)
%p A328186 local t1, t0, g, t2, n, t3;
%p A328186 if L < 2.5*M then print("Choose larger value for L");
%p A328186 else
%p A328186 t1 := 1/(1 + sin(x));
%p A328186 t0 := series(t1, x, L);
%p A328186 f := []; t2 := t0;
%p A328186 for n to M do
%p A328186 t3 := coeff(t2, x, n);
%p A328186 t2 := series(t2/(1 + t3*x^n), x, L);
%p A328186 f := [op(f), t3];
%p A328186 end do;
%p A328186 end if;
%p A328186 [seq(f[n], n = 1 .. nops(f))];
%p A328186 end proc;
%p A328186 # Calculates the denominators of f_n:
%p A328186 h := map(denom, PPE(100, 40)); # Petros Hadjicostas, Oct 06 2019 by modifying _N. J. A. Sloane_'s program from A170912 and A170913.
%t A328186 A[m_, n_] :=
%t A328186   A[m, n] =
%t A328186    Which[m == 1, 2*(-1)^n*I^(n + 2)*PolyLog[-(n + 1), -I]/n!,
%t A328186     m > n >= 1, 0, True,
%t A328186     A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
%t A328186 a[n_] := Denominator[A[n, n]];
%t A328186 a /@ Range[1, 55] (* _Petros Hadjicostas_, Oct 06 2019 using a program by _Jean-François Alcover_ and a formula from A099612 and A099617 *)
%Y A328186 Numerators are in A328191.
%Y A328186 Cf. A099612, A099617, A170914, A170915, A279107.
%K A328186 nonn,frac
%O A328186 1,3
%A A328186 _Petros Hadjicostas_, Oct 06 2019

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