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A123854
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Denominators in an asymptotic expansion for the cubic recurrence sequence A123851.
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19
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1, 4, 32, 128, 2048, 8192, 65536, 262144, 8388608, 33554432, 268435456, 1073741824, 17179869184, 68719476736, 549755813888, 2199023255552, 140737488355328, 562949953421312, 4503599627370496, 18014398509481984, 288230376151711744, 1152921504606846976
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OFFSET
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0,2
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COMMENTS
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A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Numerators are A123853.
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
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
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
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LINKS
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T. M. Apostol, On the Lerch zeta function, 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]
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FORMULA
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a(n) = denominator(binomial(1/4,n)). - Peter Luschny, Apr 07 2016
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EXAMPLE
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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.
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MAPLE
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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;
# Alternatively:
A123854 := n -> denom(binomial(1/4, n)):
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MATHEMATICA
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Denominator[CoefficientList[Series[ 1/Sqrt[Sqrt[1-x]], {x, 0, 25}], x]] (* Robert G. Wilson v, Mar 23 2014 *)
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PROG
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(PARI) vector(25, n, n--; denominator(binomial(1/4, n)) ) \\ G. C. Greubel, Aug 08 2019
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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