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

-1, 1, 7, 131, 1843, 97261, 4683059, 1331727679, 568285777, 9521655609199, 175554688130609, 11334988388673161, 3457026400678609391, 6594042537777612027841, 249248595232521829462213, 268938575250382935485761673113, 3929672369519648081411955883, 4719016202742955262333630268611

Offset: 1

Author

N. J. A. Sloane, Jan 30 2010

Keywords

• sign
• frac

Examples

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

Links

• Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
• Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
• H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
• H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
• W. Lang, Recurrences for the general problem.

Crossrefs

Cf. A170913.

Programs

Maple

t1:=cos(x);
L:=100;
t0:=series(t1,x,L):
g:=[]; M:=40; t2:=t0:
for n from 1 to M do
t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3];
od:
g;
h:=[seq(g[2*n],n=1..nops(g)/2)];
h1:=map(numer,h);
h2:=map(denom,h);

Mathematica

A[m_, n_] :=
  A[m, n] =
   Which[m == 1, (-1)^n/(2*n)!, m > n >= 1, 0, True,
    A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
a[n_] := Numerator[A[n, n]];
a /@ Range[1, 55] (* Petros Hadjicostas, Oct 04 2019, courtesy of Jean-François Alcover *)