A170912 - OEIS

%I #14 Oct 06 2019 02:38:28

%S -1,1,7,131,1843,97261,4683059,1331727679,568285777,9521655609199,

%T 175554688130609,11334988388673161,3457026400678609391,

%U 6594042537777612027841,249248595232521829462213,268938575250382935485761673113,3929672369519648081411955883,4719016202742955262333630268611

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

%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>.

%e -1/2, 1/24, 7/360, 131/13440, 1843/453600, 97261/47900160, ...

%p t1:=cos(x);

%p L:=100;

%p t0:=series(t1,x,L):

%p g:=[]; M:=40; t2:=t0:

%p for n from 1 to M do

%p t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3];

%p od:

%p g;

%p h:=[seq(g[2*n],n=1..nops(g)/2)];

%p h1:=map(numer,h);

%p h2:=map(denom,h);

%t A[m_, n_] :=

%t   A[m, n] =

%t    Which[m == 1, (-1)^n/(2*n)!, 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 04 2019, courtesy of _Jean-François Alcover_ *)

%Y Cf. A170913.

%K sign,frac

%O 1,3

%A _N. J. A. Sloane_, Jan 30 2010