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.

1, 4, -8, -16, 8992, 334144, -698752, -23349012224, 1357305243136, 6319924923392, -8773495082018816, -49004477022654464, 1709650943378038784, 480380831834367035260928, -88481173388026066736939008, -660883915180095254454665216, 888962079683152174584309088256

Offset: 0

Author

Vladimir Reshetnikov, Nov 10 2015

Keywords

• sign
• frac

Comments

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.

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).

References

• G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge, 1999; Problem 4, p. 616.
• B. C. Berndt, Ramanujan's Notebooks II, Springer, 1989; p. 181, Entry 48. See also pp. 184, 193ff.
• E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford Univ. Press, 1935; see p. 230, Problem 18.
• S. Ramanujan, Collected Papers, edited by G. H. Hardy et al., Cambridge, 1927, pp. 323-324, Question 294.

Links

• G. C. Greubel and D. Turner, Table of n, a(n) for n = 0..117
• K. P. Choi, On the medians of gamma distributions and an equation of Ramanujan, Proceedings of the American Mathematical Society 121:1 (May, 1994), pp. 245-251.
• J. C. W. Marsaglia, The incomplete gamma function and Ramanujan's rational approximation to exp(x), J. Statist. Comput. Simulation, 24 (1986), 163-168.
• Cormac O'Sullivan, Ramanujan's approximation to the exponential function and generalizations, arXiv:2205.08504 [math.NT], 2022.

Crossrefs

Cf. A065973 (denominators), A090804, A264148, A005447, A005446.

Programs

Maple

h := proc(k) option remember; local j; `if`(k<=0, 1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1),j=1..k-1))/(1+1/(k+1))) end:
A260306 := n -> `if`(n=0, 1, -numer(h(2*n+1)*doublefactorial(2*n))):
seq(A260306(n), n=0..16); # Peter Luschny, Nov 20 2015

Mathematica

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 *)

Formula

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.

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)).

Extensions

More terms from Vaclav Kotesovec, Nov 20 2015