A354173
Product_{n>=1} (1 + x^(2*n))^(a(n)/(2*n)!) = cos(x).
Original entry on oeis.org
-1, -8, 104, -12032, 354944, 47546368, 6204652544, -6174957043712, 47215125069824, 159504062197792768, 51085990673656315904, 54592541528151763714048, 15510963121850795776016384, 14479308135716773591282352128, -7469518701197092988127633473536, -77646018400552596699424746364731392
Offset: 1
-
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = {1, 0, -1, 0}[[Mod[n + 1, 4, 1]]]/n! - b[n, n - 1]; a[n_] := (2 n)! c[2 n]; Table[a[n], {n, 1, 16}]
A170916
Write sin(x)/x = Product_{n>=1} (1 + g_n*x^(2*n)); a(n) = numerator(g_n).
Original entry on oeis.org
-1, 1, 1, 73, 353, 36499, 24257, 302426881, 87721, 348958703, 226786069421, 62199570679633, 62531659610839, 8559230855533306387, 235495453816743509, 2644298730170939345197, 281737789368631676609, 39043444996461526437828311, 6203284926188598376335167
Offset: 1
-1/6, 1/120, 1/840, 73/362880, 353/14968800, 36499/9340531200, 24257/49037788800, ...
- 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.
-
t1:=sin(x)/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); # Petros Hadjicostas, Oct 04 2019 by modifying N. J. A. Sloane's program from A170912 and A170913