A170924 a(n) = numerator of the coefficient c(n) of x^n in (1/sqrt(1-x))/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

1, 3, 1, 27, 3, 39, 9, 2955, 7, 1737, 93, 88047, 315, 79779, 1083, 77010795, 3855, 488391, 13797, 905252529, 49689, 204066351, 182361, 756251509503, 10485, 10978530465, 619549, 10462007147787, 9256395, 603860858253, 34636833, 150202954242966315

Offset: 1

Author

N. J. A. Sloane, Jan 31 2010

Keywords

• nonn
• frac

Examples

			1/2, 3/8, 1/8, 27/128, 3/32, 39/512, 9/128, 2955/32768, 7/128, ...
		

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. A170923 (denominators).

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

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

Programs

Maple

L := 34: g := NULL:
t := series(1/sqrt(1 - x), x, L):
for n from 1 to L-2 do
   c := coeff(t, x, n);
   t := series(t/(1 + c*x^(n)), x, L);
   g := g, c;
od: map(numer, [g]); # Peter Luschny, May 12 2022

Extensions

Following a suggestion from Ilya Gutkovskiy, name corrected so that it matches the data by Peter Luschny, May 12 2022