A170923 a(n) = denominator of the coefficient c(n) of x^n in sqrt(1+x)/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

2, 8, 8, 128, 32, 512, 128, 32768, 128, 32768, 2048, 2097152, 8192, 2097152, 32768, 2147483648, 131072, 16777216, 524288, 34359738368, 2097152, 8589934592, 8388608, 35184372088832, 524288, 549755813888, 33554432, 562949953421312, 536870912, 35184372088832

Offset: 1

Author

N. J. A. Sloane, Jan 31 2010

Keywords

• nonn
• frac

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. A170922 (numerators).

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 := 32: g := NULL:
t := series(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(denom, [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