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 A065973 Denominators in an asymptotic expansion of Ramanujan. 7
 3, 135, 2835, 8505, 12629925, 492567075, 1477701225, 39565450299375, 2255230667064375, 6765692001193125, 7002491221234884375, 21007473663704653125, 441156946937797715625, 56995271759628775870171875 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 22 04:42 EDT 2023. Contains 365503 sequences. (Running on oeis4.)