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A090804 Numerators in an asymptotic expansion of Ramanujan.

%I A090804 #55 May 18 2022 07:53:57
%S A090804 2,-4,8,16,-8992,-334144,698752,23349012224,-1357305243136,
%T A090804 -6319924923392,8773495082018816,49004477022654464,
%U A090804 -1709650943378038784,-480380831834367035260928,88481173388026066736939008,660883915180095254454665216
%N A090804 Numerators in an asymptotic expansion of Ramanujan.
%C A090804 2^(n+1) divides a(n). - _Peter Luschny_, Nov 05 2015
%D A090804 G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge, 1999; Problem 4, p. 616.
%D A090804 B. C. Berndt, Ramanujan's Notebooks II, Springer, 1989; p. 181, Entry 48. See also pp. 184, 193ff.
%D A090804 E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford Univ. Press, 1935; see p. 230, Problem 18.
%D A090804 S. Ramanujan, Collected Papers, edited by G. H. Hardy et al., Cambridge, 1927, pp. 323-324, Question 294.
%H A090804 G. C. Greubel, <a href="/A090804/b090804.txt">Table of n, a(n) for n = 0..160</a>
%H A090804 J. D. Buckholtz, <a href="https://doi.org/10.1090/S0002-9939-1963-0151770-9">Concerning an approximation of Copson</a>, Proc. Amer. Math. Soc., 14 (1963), 564-568.
%H A090804 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 A090804 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. [From _N. J. A. Sloane_, Jun 23 2011]
%H A090804 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 A090804 Define t_n by Sum_{k=0..n-1} n^k/k! + t_n*n^n/n! = exp(n)/2; then t_n ~ 1/3 + 4/(135*n) - 8/(2835*n^2) + ...
%F A090804 Integral_{0..infinity} exp(-x)*(1+x/n)^n dx = exp(n)*Gamma(n+1)/(2*n^n) + 2/3 - 4/(135*n) + 8/(2835*n^2) + 16/(8505*n^3) - 8992/(12629925*n^4) + ...
%F A090804 From _Vladimir Reshetnikov_, Nov 09 2015: (Start)
%F A090804 Numerators/denominators: A090804(n)/A065973(n) = 0^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 * 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 A090804 A090804(n)/A065973(n) = 0^n + 2^n * (3*n+2)! * Sum_{i=1..2*n+1} Sum_{j=1..i} (-1)^j * 2^i * stirling2(2*n+i+j+1,j) / ((2*n+i+j+1)! * (2*n-i+1)! * (i-j)! * (n+i+1)).
%F A090804 (End)
%e A090804 2/3, -4/135, 8/2835, 16/8505, -8992/12629925, -334144/492567075, 698752/1477701225, ...
%p A090804 # Maple program from _N. J. A. Sloane_, Jun 23 2011, based on J. Marsaglia's 1986 paper:
%p A090804 a[1]:=1;
%p A090804 M:=20;
%p A090804 for n from 2 to M do
%p A090804 t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k],k=2..floor(n/2));
%p A090804 if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;
%p A090804 a[n]:=t1;
%p A090804 od:
%p A090804 s1:=[seq(a[n],n=1..M)]; # This gives A005447/A005446
%p A090804 s2:=[seq(2^(n+1)*(n+1)!*a[2*n+2],n=0..M/2)]; # This gives A090804/A065973
%p A090804 # minor corrections by _Peter Luschny_, Nov 05 2015
%t A090804 Numerator@Table[(n+1)! SeriesCoefficient[-(ProductLog[-1, -Exp[-x^2-1]] + ProductLog[-Exp[-x^2-1]] + 1)/(2 x^2), {x, 0, 2n}, Assumptions -> x>0], {n, 0, 15}] (* _Vladimir Reshetnikov_, Nov 09 2015 *)
%t A090804 Numerator@Table[KroneckerDelta[n] + 2^n (3n+2)! Sum[((-1)^j 2^i StirlingS2[2n+i+j+1, j])/((2n+i+j+1)! (2n-i+1)! (i-j)! (n+i+1)), {i, 1, 2n+1}, {j, 1, i}], {n, 0, 15}]  (* _Vladimir Reshetnikov_, Nov 09 2015 *)
%o A090804 (PARI) {a(n) = my(A, m); if( n<1, n==0, n++; A = vector(m=2*n, k, 1); for(k=2, m, A[k] = (A[k-1] - sum(i=2, k-1, i * A[i] * A[k+1-i])) / (k + 1)); numerator(A[m] * 2^n * n!))}; /* _Michael Somos_, Jun 09 2004 */
%Y A090804 Cf. A065973 (denominators), A005446, A005447, A264148.
%K A090804 sign,frac
%O A090804 0,1
%A A090804 _N. J. A. Sloane_, Feb 11 2004
%E A090804 a(0) corrected by _Peter Luschny_, Nov 05 2015