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A170912 Write cos(x) = Product_{n >= 1} (1 + g_n*x^(2*n)); a(n) = numerator(g_n).

%I A170912 #14 Oct 06 2019 02:38:28
%S A170912 -1,1,7,131,1843,97261,4683059,1331727679,568285777,9521655609199,
%T A170912 175554688130609,11334988388673161,3457026400678609391,
%U A170912 6594042537777612027841,249248595232521829462213,268938575250382935485761673113,3929672369519648081411955883,4719016202742955262333630268611
%N A170912 Write cos(x) = Product_{n >= 1} (1 + g_n*x^(2*n)); a(n) = numerator(g_n).
%H A170912 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 A170912 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 A170912 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 A170912 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 A170912 W. Lang, <a href="/A157162/a157162.txt">Recurrences for the general problem</a>.
%e A170912 -1/2, 1/24, 7/360, 131/13440, 1843/453600, 97261/47900160, ...
%p A170912 t1:=cos(x);
%p A170912 L:=100;
%p A170912 t0:=series(t1,x,L):
%p A170912 g:=[]; M:=40; t2:=t0:
%p A170912 for n from 1 to M do
%p A170912 t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3];
%p A170912 od:
%p A170912 g;
%p A170912 h:=[seq(g[2*n],n=1..nops(g)/2)];
%p A170912 h1:=map(numer,h);
%p A170912 h2:=map(denom,h);
%t A170912 A[m_, n_] :=
%t A170912   A[m, n] =
%t A170912    Which[m == 1, (-1)^n/(2*n)!, m > n >= 1, 0, True,
%t A170912     A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
%t A170912 a[n_] := Numerator[A[n, n]];
%t A170912 a /@ Range[1, 55] (* _Petros Hadjicostas_, Oct 04 2019, courtesy of _Jean-François Alcover_ *)
%Y A170912 Cf. A170913.
%K A170912 sign,frac
%O A170912 1,3
%A A170912 _N. J. A. Sloane_, Jan 30 2010