cp's OEIS Frontend

A116603 Coefficients in asymptotic expansion of sequence A052129.

%I A116603 #76 May 27 2025 10:10:21
%S A116603 1,2,-1,4,-21,138,-1091,10088,-106918,1279220,-17070418,251560472,
%T A116603 -4059954946,71250808916,-1351381762990,27552372478592,
%U A116603 -601021307680207,13969016314470386,-344653640328891233,8997206549370634644,-247772400254700937149,7178881153198162073002
%N A116603 Coefficients in asymptotic expansion of sequence A052129.
%D A116603 Steven R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
%H A116603 Chao-Ping Chen, <a href="https://doi.org/10.1016/j.jnt.2016.08.010">Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant</a>, Journal of Number Theory, 172 (2017), 145-159.
%H A116603 Chao-Ping Chen and X.-F. Han, <a href="http://dx.doi.org/10.1016/j.jnt.2016.02.018">On Somos' quadratic recurrence constant</a>, Journal of Number Theory, 166 (2016), 31-40.
%H A116603 Dawei Lu and Zexi Song, <a href="https://doi.org/10.1016/j.jnt.2015.03.013">Some new continued fraction estimates of the Somos' quadratic recurrence constant</a>, Journal of Number Theory, 155 (2015), 36-45.
%H A116603 Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, <a href="https://doi.org/10.1007/s00025-018-0928-0">Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant</a>, Results in Mathematics 74(1) (2019), Article no. 6.
%H A116603 Gergo Nemes, <a href="http://www.jstor.org/stable/43666828">On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant</a>, Applicable Analysis and Discrete Mathematics, 5(1) (2011), 60-66.
%H A116603 Jörg Neunhäuserer, <a href="https://arxiv.org/abs/2006.02882">On the universality of Somos' constant</a>, arXiv:2006.02882 [math.DS], 2020.
%H A116603 Jonathan Sondow and Petros Hadjicostas, <a href="http://arxiv.org/abs/math/0610499">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, arXiv:0610499 [math.CA], 2006.
%H A116603 Jonathan Sondow and Petros Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl. 332 (2007), 292-314.
%H A116603 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html">Somos's Quadratic Recurrence Constant</a>.
%H A116603 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoebelsSequence.html">Goebel's Sequence</a>.
%H A116603 Xu You and Di-Rong Chen, <a href="https://doi.org/10.1016/j.jmaa.2015.12.013">Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant</a>, Mathematical Analysis and Applications, 436(1) (2016), 513-520.
%H A116603 Aimin Xu, <a href="https://doi.org/10.1186/s13660-019-2153-0">Approximations of the generalized-Euler-constant function and the generalized Somos' quadratic recurrence constant</a>, Journal of Inequalities and Applications, Vol. 2019 (2019), Article No. 198.
%F A116603 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
%F A116603 G.f. A(x) satisfies (1 + x)^2 = A(x)^2 / A(x/(1 + x)).
%F A116603 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).
%F A116603 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).
%F A116603 From _Seiichi Manyama_, May 26 2025: (Start)
%F A116603 G.f.: Product_{k>=1} (1 + k*x)^(1/2^k).
%F A116603 G.f.: exp(2 * Sum_{k>=1} (-1)^(k-1) * A000670(k) * x^k/k).
%F A116603 G.f.: 1/B(-x), where B(x) is the g.f. of A084785. (End)
%F A116603 a(n) ~ (-1)^(n+1) * (n-1)! / log(2)^(n+1). - _Vaclav Kotesovec_, May 27 2025
%e A116603 G.f. = 1 + 2*x - x^2 + 4*x^3 - 21*x^4 + 138*x^5 - 1091*x^6 + 10088*x^7 + ...
%t A116603 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 *)
%o A116603 (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))};
%Y A116603 Cf. A052129, A112302, A123851, A123852, A123853, A123854.
%Y A116603 Cf. A000670, A084785.
%K A116603 sign
%O A116603 0,2
%A A116603 _Michael Somos_, Feb 18 2006