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%I #57 Aug 02 2023 11:43:50
%S 1,4,32,128,2048,8192,65536,262144,8388608,33554432,268435456,
%T 1073741824,17179869184,68719476736,549755813888,2199023255552,
%U 140737488355328,562949953421312,4503599627370496,18014398509481984,288230376151711744,1152921504606846976
%N Denominators in an asymptotic expansion for the cubic recurrence sequence A123851.
%C A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Numerators are A123853.
%C Equals 2^A004134(n); also the denominators in expansion of (1-x)^(-1/4). - _Alexander Adamchuk_, Oct 27 2006
%C All terms are powers of 2 and log_2 a(n) = A004134(n) = 3*n - A000120(n). - _Alexander Adamchuk_, Oct 27 2006 [Edited by _Petros Hadjicostas_, May 14 2020]
%C Is this the same sequence as A088802? - _N. J. A. Sloane_, Mar 21 2007
%C Almost certainly this is the same as A088802. - _Michael Somos_, Aug 23 2007
%C Denominators of Gegenbauer_C(2n,1/4,2). The denominators of Gegenbauer_C(n,1/4,2) give the doubled sequence. - _Paul Barry_, Apr 21 2009
%C If the Greubel formula in A088802 and the Luschny formula here are correct (they are the same), the sequence is a duplicate of A088802. - _R. J. Mathar_, Aug 02 2023
%D S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
%H G. C. Greubel, <a href="/A123854/b123854.txt">Table of n, a(n) for n = 0..1000</a>
%H T. M. Apostol, <a href="https://projecteuclid.org/euclid.pjm/1103052188">On the Lerch zeta function</a>, Pacific J. Math. 1 (1951), 161-167. [In Eq. (3.7), p. 166, the index in the summation for the Apostol-Bernoulli numbers should start at s = 0, not at s = 1. - _Petros Hadjicostas_, Aug 09 2019]
%H Jonathan Sondow and Petros Hadjicostas, <a href="http://arXiv.org/abs/math/0610499">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, arXiv:math/0610499 [math.CA], 2006.
%H Jonathan Sondow and Petros Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl. 332 (2007), 292-314.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html">Somos's Quadratic Recurrence Constant</a>.
%H Aimin Xu, <a href="https://doi.org/10.1142/S1793042119501112">Asymptotic expansion related to the Generalized Somos Recurrence constant</a>, International Journal of Number Theory 15(10) (2019), 2043-2055. [The author gives recurrences and other formulas for the coefficients of the asymptotic expansion using the Apostol-Bernoulli numbers (see the reference above) and the Bell polynomials. - _Petros Hadjicostas_, Aug 09 2019]
%F From _Alexander Adamchuk_, Oct 27 2006: (Start)
%F a(n) = 2^A004134(n).
%F a(n) = 2^(3n - A000120(n)). (End)
%F a(n) = denominator(binomial(1/4,n)). - _Peter Luschny_, Apr 07 2016
%e A123851(n) ~ c^(3^n)*n^(- 1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) - 5397/(2048*n^4) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.
%p f:=proc(t,x) exp(sum(ln(1+m*x)/t^m,m=1..infinity)); end; for j from 0 to 29 do denom(coeff(series(f(3,x),x=0,30),x,j)); od;
%p # Alternatively:
%p A123854 := n -> denom(binomial(1/4,n)):
%p seq(A123854(n), n=0..25); # _Peter Luschny_, Apr 07 2016
%t Denominator[CoefficientList[Series[ 1/Sqrt[Sqrt[1-x]], {x, 0, 25}], x]] (* _Robert G. Wilson v_, Mar 23 2014 *)
%o (Sage) # uses[A000120]
%o def A123854(n): return 1 << (3*n-A000120(n))
%o [A123854(n) for n in (0..25)] # _Peter Luschny_, Dec 02 2012
%o (PARI) vector(25, n, n--; denominator(binomial(1/4,n)) ) \\ _G. C. Greubel_, Aug 08 2019
%Y Cf. A052129, A112302, A116603, A123851, A123852, A123853 (numerators).
%Y Cf. A004134, A004130, A000120.
%K frac,nonn
%O 0,2
%A _Petros Hadjicostas_ and _Jonathan Sondow_, Oct 15 2006
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