A065973 Denominators in an asymptotic expansion of Ramanujan.

3, 135, 2835, 8505, 12629925, 492567075, 1477701225, 39565450299375, 2255230667064375, 6765692001193125, 7002491221234884375, 21007473663704653125, 441156946937797715625, 56995271759628775870171875

Offset

0,1

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

Robert Israel, Table of n, a(n) for n = 0..320 (0 .. 126 from G. C. Greubel and D. Turner)

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.

Formula

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) + ...

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) + ...

Example

-2/3, 4/135, -8/2835, -16/8505, 8992/12629925, 334144/492567075, -698752/1477701225, ...

Maple

# Maple program from N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper:
a[1]:=1;
M:=20;
for n from 2 to M do
t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k], k=2..floor(n/2));
if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;
a[n]:=t1;
od:
s1:=[seq(a[n], n=1..M)]: # This gives A005447/A005446
s2:=[seq(-2^(n+1)*(n+1)!*a[2*n+2], n=0..(M-2)/2)]: # This gives A090804/A065973
map(denom, s2);

Mathematica

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 *)

Prog

(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 */

Crossrefs

Cf. A260306 (numerators), A090804, A005446, A005447.

Sequence in context: A051376 A101721 A173582 * A110973 A361195 A136411

Adjacent sequences: A065970 A065971 A065972 * A065974 A065975 A065976

Keyword

nonn,frac

Author

N. J. A. Sloane, Dec 09 2001

Extensions

Maple program edited by Robert Israel, Dec 15 2015

Status

approved