cp's OEIS Frontend

A123853 Numerators in an asymptotic expansion for the cubic recurrence sequence A123851.

%I A123853 #38 Feb 16 2025 08:33:03
%S A123853 1,3,-15,113,-5397,84813,-3267755,74391561,-15633072909,465681118929,
%T A123853 -31041303829713,1145088996404679,-185348722911971841,
%U A123853 8165727090278785521,-778296382754673737187,39898888480559205453945,-35033447016186321707305533
%N A123853 Numerators in an asymptotic expansion for the cubic recurrence sequence A123851.
%C A123853 A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Denominators are A123854.
%D A123853 Steven R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
%H A123853 T. M. Apostol, <a href="https://projecteuclid.org/euclid.pjm/1103052188">On the Lerch zeta function</a>, 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]
%H A123853 Jonathan Sondow and Petros Hadjicostas, <a href="https://arxiv.org/abs/math/0610499">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, arXiv:math/0610499 [math.CA], 2006.
%H A123853 Jonathan Sondow and Petros Hadjicostas, <a href="https://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 A123853 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html">Somos's Quadratic Recurrence Constant</a>.
%H A123853 Aimin Xu, <a href="https://doi.org/10.1142/S1793042119501112">Asymptotic expansion related to the Generalized Somos Recurrence constant</a>, International Journal of Number Theory 15(10) (2019), 2043-2055. [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]
%e A123853 A123851(n) ~ c^(3^n)*n^(-1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) - 5397/(2048*n^4) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.
%p A123853 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;
%o A123853 (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 */
%Y A123853 Cf. A052129, A112302, A116603, A123851, A123852, A123854 (denominators).
%K A123853 frac,sign
%O A123853 0,2
%A A123853 _Petros Hadjicostas_ and _Jonathan Sondow_, Oct 15 2006