A170918 a(n) = numerator of the coefficient c(n) of x^n in (tan x)/Product_{0 < k < n} 1 + c(k)*x^k, n = 1, 2, 3, ...
1, -1, 7, -14, 54, -1112, 6574, -48488, 1143731, -14813072, 16252211, -3500388967, 125127865048, -158589803803, 33133618166566, -30512906279732, 4378989933312913, -330336346477870319, 1981395373839282068, -251479418962683770473, 79893293800974935213, -31493610597939643431532
Offset: 1
Examples
1, -1, 7/3, -14/3, 54/5, -1112/45, 6574/105, -48488/315, 1143731/2835, ...
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.
- Wolfdieter Lang, Recurrences for the general problem.
Crossrefs
Programs
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Maple
t1:=tan(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; g1:=map(numer,g); g2:=map(denom,g);
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PARI
t=tan(x+O(x)^25); vector(#t,n,c=polcoef(t,n);t/=1+c*x^n;numerator(c)) \\ M. F. Hasler, May 07 2022
Extensions
Following a suggestion from Ilya Gutkovskiy, name corrected so that it matches the data by M. F. Hasler, May 07 2022