A353583 Numerators of coefficients c(n) in product expansion of 1 + tan x = Product_{k>=1} 1 + c(k)*x^k.

1, 0, 1, -1, 7, -7, 199, -71, 484, -368, 187909, -610103, 2068657, -63614, 1530164189, -1715846683, 7628902283, -125125345078, 9521826231889, -17921564328719, 291162274608871, -47147385565688, 552647133893696333, -36898601487519532, 4761064630028162378

Offset: 1

Author

M. F. Hasler, May 07 2022

Keywords

• sign
• frac

Comments

See A353584 for the denominators, and A353586 for the analog for (tan x)/x.

Examples

			1 + tan x = (1 + x)(1 + 1/3*x^3)(1 - 1/3*x^4/3)(1 + 7/15*x^5)(1 - 7/15*x^5)(...),
and this sequence lists the numerators of (1, 0, 1/3, -1/3, 7/15, -7/15, ...).
		

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

Cf. A353584 (denominators), A353586 / A353587 (similar for (tan x)/x).

Cf. A170918 / A170919 for a variant.

Programs

PARI

t=1+tan(x+O(x)^29); vector(#t-1,n,c=polcoef(t,n);t/=1+c*x^n;numerator(c))