A170923 -id:A170923 - cp's OEIS frontend

A170922 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, ...

1, -1, 1, -13, 3, -37, 9, -1861, 7, -1491, 93, -81001, 315, -69705, 1083, -63586357, 3855, -438821, 13797, -822684711, 49689, -186369117, 182361, -704368012465, 10485, -10165801275, 619549, -9738266477517, 9256395, -566066862375, 34636833, -140047960975823893

Offset: 1

N. J. A. Sloane, Jan 31 2010

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

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.

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.

L := 34: 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(numer, [g]); # Peter Luschny, May 12 2022

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

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, ...

Original entry on oeis.org

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

N. J. A. Sloane, Jan 31 2010

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

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.

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.

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

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

cp's OEIS Frontend

A170922 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, ...

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