A170922 - OEIS

%I #19 Jul 23 2023 18:23:46

%S 1,-1,1,-13,3,-37,9,-1861,7,-1491,93,-81001,315,-69705,1083,-63586357,

%T 3855,-438821,13797,-822684711,49689,-186369117,182361,-704368012465,

%U 10485,-10165801275,619549,-9738266477517,9256395,-566066862375,34636833,-140047960975823893

%N a(n) = numerator 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, ...

%H Giedrius Alkauskas, <a href="http://arxiv.org/abs/0801.0805">One curious proof of Fermat's little theorem</a>, arXiv:0801.0805 [math.NT], 2008.

%H Giedrius Alkauskas, <a href="https://www.jstor.org/stable/40391097">A curious proof of Fermat's little theorem</a>, Amer. Math. Monthly 116(4) (2009), 362-364.

%H Giedrius Alkauskas, <a href="https://arxiv.org/abs/1609.09842">Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems</a>, arXiv:1609.09842 [math.NT], 2016.

%H H. Gingold, H. W. Gould, and Michael E. Mays, <a href="https://www.researchgate.net/publication/268023169_Power_product_expansions">Power Product Expansions</a>, Utilitas Mathematica 34 (1988), 143-161.

%H H. Gingold and A. Knopfmacher, <a href="http://dx.doi.org/10.4153/CJM-1995-062-9">Analytic properties of power product expansions</a>, Canad. J. Math. 47 (1995), 1219-1239.

%H Wolfdieter Lang, <a href="/A157162/a157162.txt">Recurrences for the general problem</a>.

%e 1/2, -1/8, 1/8, -13/128, 3/32, -37/512, 9/128, -1861/32768, ...

%p L := 34: g := NULL:

%p t := series(sqrt(1+x), x, L):

%p for n from 1 to L-2 do

%p    c := coeff(t, x, n);

%p    t := series(t/(1 + c*x^n), x, L);

%p    g := g, c;

%p od: map(numer, [g]); # _Peter Luschny_, May 12 2022

%Y Cf. A170923 (denominators).

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

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

%K sign,frac

%O 1,4

%A _N. J. A. Sloane_, Jan 31 2010

%E Following a suggestion from _Ilya Gutkovskiy_, name corrected so that it matches the data by _Peter Luschny_, May 12 2022