login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A123853 Numerators in asymptotic expansion of cubic recurrence sequence A123851. 6
1, 3, -15, 113, -5397, 84813, -3267755, 74391561, -15633072909, 465681118929, -31041303829713, 1145088996404679, -185348722911971841, 8165727090278785521, -778296382754673737187, 39898888480559205453945, -35033447016186321707305533 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Denominators are A123854.

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

LINKS

Table of n, a(n) for n=0..16.

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]

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.

Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.

Aimin Xu, Asymptotic expansion related to the Generalized Somos Recurrence constant, International Journal of Number Theory (2019), to appear. [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]

EXAMPLE

A123851(n) ~ c^(3^n)*n^(-1/2)/(1 + 3/4n - 15/32n^2 + 113/128n^3 - 5397/2048n^4 + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.

MAPLE

f:=proc(t, x) exp(sum(ln(1+m*x)/t^m, m=1..infinity)); end; for j from 0 to 29 do numer(coeff(series(f(3, x), x=0, 30), x, j)); od;

PROG

(PARI) {a(n) = local(A); if(n < 0, 0, A = 1 + O(x) ; for( k = 1, n, A = truncate(A) + x * O(x^k); A += x^k * polcoeff( 3/4 * (subst(1/A, x, x^2/(1-x^2))^2/(1-x^2) - 1/subst(A, x, x^2)^(2/3)), 2*k ) ); numerator( polcoeff( A, n ) ) ) } /* Michael Somos, Aug 23 2007 */

CROSSREFS

Cf. A052129, A112302, A116603, A123851, A123852, A123854.

Sequence in context: A056053 A295758 A059849 * A166885 A082163 A331122

Adjacent sequences:  A123850 A123851 A123852 * A123854 A123855 A123856

KEYWORD

frac,sign

AUTHOR

Petros Hadjicostas and Jonathan Sondow, Oct 15 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 18 04:48 EST 2020. Contains 332011 sequences. (Running on oeis4.)