A170917 Write sin(x)/x = Product_{n>=1} (1 + g_n*x^(2*n)); a(n) = denominator(g_n).
6, 120, 840, 362880, 14968800, 9340531200, 49037788800, 3201186852864000, 8485288812000, 182467650613248000, 908859963476424960000, 1424498881530396672000000, 10633661572674172032000000, 8289151869130970582384640000000, 1720739115690134518218240000000, 97858575719142221963014963200000000
Offset: 1
Examples
-1/6, 1/120, 1/840, 73/362880, 353/14968800, 36499/9340531200, 24257/49037788800, ...
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.
- Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.
- 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.
Programs
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Maple
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