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A116603 Coefficients in asymptotic expansion of sequence A052129. 7
1, 2, -1, 4, -21, 138, -1091, 10088, -106918, 1279220, -17070418, 251560472, -4059954946, 71250808916, -1351381762990, 27552372478592, -601021307680207, 13969016314470386, -344653640328891233, 8997206549370634644 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

LINKS

Table of n, a(n) for n=0..19.

Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 2016, Volume 172, March 2017, Pages 145-159; https://doi.org/10.1016/j.jnt.2016.08.010

Chao-Ping Chen, X.-F. Han, On Somos' quadratic recurrence constant, Journal of Number Theory, Volume 166, September 2016, Pages 31-40.

Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, Volume 155, October 2015, Pages 36-45.

Dawei Lu, Xiaoguang Wang, Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics (2019) Vol. 74, No. 1, 6.

Gergo Nemes, On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant, Applicable Analysis and Discrete Mathematics, Vol. 5, No. 1 (April 2011), pp. 60-66.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv: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

Eric Weisstein's World of Mathematics, Goebel's Sequence

Xu You, Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, Volume 436, Issue 1, 1 April 2016, Pages 513-520.

Aimin Xu, Approximations of the generalized-Euler-constant function and the generalized Somos' quadratic recurrence constant, Journal of Inequalities and Applications (2019) Vol. 2019, Article No. 198.

FORMULA

a(0)=1; thereafter, a(n) = (1/n)*Sum((-1)^(j-1)*2*b(j)*a(n-j),j=1..n) where b(j) = A000670(j) [Nemes]. - N. J. A. Sloane, Sep 11 2017

G.f. A(x) satisfies (1 + x)^2 = A(x)^2 / A(x/(1 + x)).

A003504(n+1) ~ C^(2^n) * (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...) where C = 1.04783144757... (see A115632).

A052129(n) ~ s^(2^n) / (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...) where s = 1.661687949633... (see A112302).

EXAMPLE

G.f. = 1 + 2*x - x^2 + 4*x^3 - 21*x^4 + 138*x^5 - 1091*x^6 + 10088*x^7 + ...

MATHEMATICA

terms = 20; A[_] = 1; Do[A[x_] = -A[x] + 2/A[x/(1+x)]^(-1/2)*(1+x) + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Jean-Fran├žois Alcover, Jul 28 2011, updated Jan 12 2018 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A=1; for( k=1, n, A = truncate( A + O(x^k)) + x * O(x^k); A = -A + 2 / subst(A^(-1/2), x, x/(1 + x)) * (1 + x); ); polcoeff(A, n))};

CROSSREFS

Cf. A052129, A112302, A123851, A123852, A123853, A123854.

Sequence in context: A032105 A259472 A053565 * A158356 A015939 A261059

Adjacent sequences:  A116600 A116601 A116602 * A116604 A116605 A116606

KEYWORD

sign

AUTHOR

Michael Somos, Feb 18 2006

STATUS

approved

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Last modified October 23 21:57 EDT 2019. Contains 328373 sequences. (Running on oeis4.)