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

1,3

Links

Table of n, a(n) for n=1..22.

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.

Example

1, -1, 7/3, -14/3, 54/5, -1112/45, 6574/105, -48488/315, 1143731/2835, ...

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);

Prog

(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

Crossrefs

Cf. A170919 (denominators), A170910-A170917.

Cf. A353583 / A353584 for power product expansion of 1 + tan x.

Cf. A353586 / A353587 for power product expansion of (tan x)/x.

Sequence in context: A045759 A166637 A237686 * A033650 A135536 A241201

Adjacent sequences: A170915 A170916 A170917 * A170919 A170920 A170921

Keyword

sign,frac

Author

N. J. A. Sloane, Jan 30 2010

Extensions

Following a suggestion from Ilya Gutkovskiy, name corrected so that it matches the data by M. F. Hasler, May 07 2022

Status

approved