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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.

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.

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 December 15 08:47 EST 2018. Contains 318147 sequences. (Running on oeis4.)