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Search: id:A090804
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| A090804 |
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Numerators in an asymptotic expansion of Ramanujan. |
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+0
2
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| 1, -4, 8, 16, -8992, -334144, 698752, 23349012224, -1357305243136, -6319924923392, 8773495082018816, 49004477022654464, -1709650943378038784, -480380831834367035260928, 88481173388026066736939008, 660883915180095254454665216
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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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.
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FORMULA
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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) + ...
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PROGRAM
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(PARI) a(n)=local(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 */
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CROSSREFS
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Cf. A065973.
Sequence in context: A023376 A038110 A130436 this_sequence A074025 A031462 A045066
Adjacent sequences: A090801 A090802 A090803 this_sequence A090805 A090806 A090807
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KEYWORD
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sign,frac
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AUTHOR
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njas, Feb 11, 2004
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