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A065973
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Denominators in an asymptotic expansion of Ramanujan.
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7
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3, 135, 2835, 8505, 12629925, 492567075, 1477701225, 39565450299375, 2255230667064375, 6765692001193125, 7002491221234884375, 21007473663704653125, 441156946937797715625, 56995271759628775870171875
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OFFSET
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0,1
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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.
S. Ramanujan, Collected Papers, edited by G. H. Hardy et al., Cambridge, 1927, pp. 323-324, Question 294.
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LINKS
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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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EXAMPLE
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-2/3, 4/135, -8/2835, -16/8505, 8992/12629925, 334144/492567075, -698752/1477701225, ...
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MAPLE
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# 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:
s2:=[seq(-2^(n+1)*(n+1)!*a[2*n+2], n=0..(M-2)/2)]: # This gives A090804/A065973
map(denom, s2);
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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