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A260306 Numerators in Ramanujan's asymptotic expansion of theta(n), defined by Sum_{k=0..n-1} n^k/k! + theta(n)*n^n/n! = exp(n)/2.

%I A260306 #24 May 18 2022 07:53:53
%S A260306 1,4,-8,-16,8992,334144,-698752,-23349012224,1357305243136,
%T A260306 6319924923392,-8773495082018816,-49004477022654464,
%U A260306 1709650943378038784,480380831834367035260928,-88481173388026066736939008,-660883915180095254454665216,888962079683152174584309088256
%N A260306 Numerators in Ramanujan's asymptotic expansion of theta(n), defined by Sum_{k=0..n-1} n^k/k! + theta(n)*n^n/n! = exp(n)/2.
%C A260306 Let Sum_{k=0..n-1} n^k/k! + theta(n)*n^n/n! = exp(n)/2. Ramanujan gave initial terms of the asymptotic expansion theta(n) = 1/3 + (4/135)/n + (-8/2835)/n^2 + (-16/8505)/n^3 + O(1/n^4). This sequence gives the numerators in this expansion, and A065973 gives the denominators.
%C A260306 Ramanujan's asymptotic is also considered in the literature in the form 1-theta(n) (see for example formula (5) in the Choi link). The numerators that appear in that form are given in A090804 (the denominators are the same). The first term A090804(0) = 2 is different, and signs of other terms are opposite to a(n).
%D A260306 G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge, 1999; Problem 4, p. 616.
%D A260306 B. C. Berndt, Ramanujan's Notebooks II, Springer, 1989; p. 181, Entry 48. See also pp. 184, 193ff.
%D A260306 E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford Univ. Press, 1935; see p. 230, Problem 18.
%D A260306 S. Ramanujan, Collected Papers, edited by G. H. Hardy et al., Cambridge, 1927, pp. 323-324, Question 294.
%H A260306 G. C. Greubel and D. Turner, <a href="/A260306/b260306.txt">Table of n, a(n) for n = 0..117</a>
%H A260306 K. P. Choi, <a href="http://dx.doi.org/10.1090/S0002-9939-1994-1195477-8">On the medians of gamma distributions and an equation of Ramanujan</a>, Proceedings of the American Mathematical Society 121:1 (May, 1994), pp. 245-251.
%H A260306 J. C. W. Marsaglia, <a href="https://doi.org/10.1080/00949658608810899">The incomplete gamma function and Ramanujan's rational approximation to exp(x)</a>, J. Statist. Comput. Simulation, 24 (1986), 163-168.
%H A260306 Cormac O'Sullivan, <a href="https://arxiv.org/abs/2205.08504">Ramanujan's approximation to the exponential function and generalizations</a>, arXiv:2205.08504 [math.NT], 2022.
%F A260306 Numerators/denominators: a(n)/A065973(n) = 2^n * (3*n+2)! / (2*n+1)! * Sum_{i=1..2*n+1} Sum_{j=1..i} Sum_{k=1..j} (-1)^(k+1) * 2^i * k^(2*n+i+j+1) * C(2*n+1,i) * C(i,j) * C(j,k) / ((2*n+i+j+1)! * (n+i+1)), where C(n,k) = A007318(n,k) are binomial coefficients.
%F A260306 a(n)/A065973(n) = 2^n * (3*n+2)! * Sum_{i=1..2*n+1} Sum_{j=1..i} (-1)^(j+1) * 2^i * stirling2(2*n+i+j+1,j) / ((2*n+i+j+1)! * (2*n-i+1)! * (i-j)! * (n+i+1)).
%p A260306 h := proc(k) option remember; local j; `if`(k<=0, 1,
%p A260306 (h(k-1)/k-add((h(k-j)*h(j))/(j+1),j=1..k-1))/(1+1/(k+1))) end:
%p A260306 A260306 := n -> `if`(n=0, 1, -numer(h(2*n+1)*doublefactorial(2*n))):
%p A260306 seq(A260306(n), n=0..16); # _Peter Luschny_, Nov 20 2015
%t A260306 Numerator[Table[2^n*(3*n + 2)! * Sum[ Sum[ (-1)^(j + 1)*2^i*StirlingS2[2*n + i + j + 1, j]/((2*n + i + j + 1)!*(2*n - i + 1)!*(i - j)!*(n + i + 1)), {j, 1, i}], {i, 1, 2*n+1}], {n, 0, 20}]] (* _Vaclav Kotesovec_, Nov 20 2015 *)
%Y A260306 Cf. A065973 (denominators), A090804, A264148, A005447, A005446.
%K A260306 sign,frac
%O A260306 0,2
%A A260306 _Vladimir Reshetnikov_, Nov 10 2015
%E A260306 More terms from _Vaclav Kotesovec_, Nov 20 2015