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A065973 Denominators in an asymptotic expansion of Ramanujan.

%I A065973 #38 May 18 2022 07:53:49
%S A065973 3,135,2835,8505,12629925,492567075,1477701225,39565450299375,
%T A065973 2255230667064375,6765692001193125,7002491221234884375,
%U A065973 21007473663704653125,441156946937797715625,56995271759628775870171875
%N A065973 Denominators in an asymptotic expansion of Ramanujan.
%D A065973 G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge, 1999; Problem 4, p. 616.
%D A065973 B. C. Berndt, Ramanujan's Notebooks II, Springer, 1989; p. 181, Entry 48. See also pp. 184, 193ff.
%D A065973 E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford Univ. Press, 1935; see p. 230, Problem 18.
%D A065973 S. Ramanujan, Collected Papers, edited by G. H. Hardy et al., Cambridge, 1927, pp. 323-324, Question 294.
%H A065973 Robert Israel, <a href="/A065973/b065973.txt">Table of n, a(n) for n = 0..320</a> (0 .. 126 from G. C. Greubel and D. Turner)
%H A065973 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 A065973 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 A065973 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 A065973 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) + ...
%e A065973 -2/3, 4/135, -8/2835, -16/8505, 8992/12629925, 334144/492567075, -698752/1477701225, ...
%p A065973 # Maple program from _N. J. A. Sloane_, Jun 23 2011, based on J. Marsaglia's 1986 paper:
%p A065973 a[1]:=1;
%p A065973 M:=20;
%p A065973 for n from 2 to M do
%p A065973 t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k],k=2..floor(n/2));
%p A065973 if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;
%p A065973 a[n]:=t1;
%p A065973 od:
%p A065973 s1:=[seq(a[n],n=1..M)]: # This gives A005447/A005446
%p A065973 s2:=[seq(-2^(n+1)*(n+1)!*a[2*n+2],n=0..(M-2)/2)]: # This gives A090804/A065973
%p A065973 map(denom,s2);
%t A065973 Denominator[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 *)
%o A065973 (PARI) a(n)=local(A,m); if(n<0,0,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)); denominator(A[m]*2^n*n!)) /* _Michael Somos_, Jun 09 2004 */
%Y A065973 Cf. A260306 (numerators), A090804, A005446, A005447.
%K A065973 nonn,frac
%O A065973 0,1
%A A065973 _N. J. A. Sloane_, Dec 09 2001
%E A065973 Maple program edited by _Robert Israel_, Dec 15 2015