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A170913 Write cos(x) = Product_{n>=1} (1 + g_n*x^(2*n)); a(n) = denominator(g_n).

%I A170913 #20 Oct 06 2019 02:38:18
%S A170913 2,24,360,13440,453600,47900160,5448643200,2988969984000,
%T A170913 3126159036000,101370917007360000,4390627842881280000,
%U A170913 552984315270266880000,393839317506450816000000,1465809349094778175488000000,129517997955171415349760000000,263130836933693530167218012160000000
%N A170913 Write cos(x) = Product_{n>=1} (1 + g_n*x^(2*n)); a(n) = denominator(g_n).
%H A170913 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 A170913 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 A170913 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 A170913 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 A170913 W. Lang, <a href="/A157162/a157162.txt">Recurrences for the general problem</a>.
%e A170913 -1/2, 1/24, 7/360, 131/13440, 1843/453600, 97261/47900160, ...
%p A170913 t1:=cos(x);
%p A170913 L:=100;
%p A170913 t0:=series(t1, x, L):
%p A170913 g:=[]; M:=40; t2:=t0:
%p A170913 for n from 1 to M do
%p A170913 t3:=coeff(t2, x, n); t2:=series(t2/(1+t3*x^n), x, L); g:=[op(g), t3];
%p A170913 od:
%p A170913 g;
%p A170913 h:=[seq(g[2*n], n=1..nops(g)/2)];
%p A170913 h1:=map(numer, h);
%p A170913 h2:=map(denom, h);
%t A170913 A[m_, n_] :=
%t A170913   A[m, n] =
%t A170913    Which[m == 1, (-1)^n/(2*n)!, m > n >= 1, 0, True,
%t A170913     A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
%t A170913 a[n_] := Denominator[A[n, n]];
%t A170913 a /@ Range[1, 55] (* _Petros Hadjicostas_, Oct 04 2019, courtesy of  _Jean-François Alcover_ *)
%Y A170913 Cf. A170912.
%K A170913 nonn,frac
%O A170913 1,1
%A A170913 _N. J. A. Sloane_, Jan 30 2010