A170913 - OEIS

%I #22 May 30 2026 16:40:33

%S 2,24,360,13440,453600,47900160,5448643200,2988969984000,

%T 3126159036000,101370917007360000,4390627842881280000,

%U 552984315270266880000,393839317506450816000000,1465809349094778175488000000,129517997955171415349760000000,263130836933693530167218012160000000

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

%H Giedrius Alkauskas, <a href="https://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="https://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_] := Denominator[A[n, n]];

%t a /@ Range[1, 55] (* _Petros Hadjicostas_, Oct 04 2019, courtesy of  _Jean-François Alcover_ *)

%Y Cf. A170912.

%K nonn,frac,changed

%O 1,1

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