A116603 Coefficients in asymptotic expansion of sequence A052129.

1, 2, -1, 4, -21, 138, -1091, 10088, -106918, 1279220, -17070418, 251560472, -4059954946, 71250808916, -1351381762990, 27552372478592, -601021307680207, 13969016314470386, -344653640328891233, 8997206549370634644

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, 172 (2017), 145-159.

Chao-Ping Chen and X.-F. Han, On Somos' quadratic recurrence constant, Journal of Number Theory, 166 (2016), 31-40.

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

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

Gergo Nemes, On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant, Applicable Analysis and Discrete Mathematics, 5(1) (2011), 60-66.

Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.

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

Jonathan Sondow and Petros 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 and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.

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

Formula

a(0) = 1; thereafter, a(n) = (1/n)*Sum_{j=1..n} (-1)^(j-1)*2*b(j)*a(n-j), 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-François 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: A259472 A354055 A053565 * A354056 A158356 A015939

Adjacent sequences: A116600 A116601 A116602 * A116604 A116605 A116606

Keyword

sign

Author

Michael Somos, Feb 18 2006

Status

approved