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A090804
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Numerators in an asymptotic expansion of Ramanujan.
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6
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2, -4, 8, 16, -8992, -334144, 698752, 23349012224, -1357305243136, -6319924923392, 8773495082018816, 49004477022654464, -1709650943378038784, -480380831834367035260928, 88481173388026066736939008, 660883915180095254454665216
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history;
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OFFSET
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0,1
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COMMENTS
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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) + ...
Numerators/denominators: A090804(n)/A065973(n) = 0^n + 2^n * (3*n+2)! / (2*n+1)! * Sum_{i=1..2*n+1} Sum_{j=1..i} Sum_{k=1..j} (-1)^k * 2^i * k^(2*n+i+j+1) * C(2*n+1,i) * C(i,j) * C(j,k) / ((2*n+i+j+1)! * (n+i+1)), where C(n,k) = A007318(n,k) are binomial coefficients.
A090804(n)/A065973(n) = 0^n + 2^n * (3*n+2)! * Sum_{i=1..2*n+1} Sum_{j=1..i} (-1)^j * 2^i * stirling2(2*n+i+j+1,j) / ((2*n+i+j+1)! * (2*n-i+1)! * (i-j)! * (n+i+1)).
(End)
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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)]; # This gives A090804/A065973
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MATHEMATICA
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Numerator@Table[(n+1)! SeriesCoefficient[-(ProductLog[-1, -Exp[-x^2-1]] + ProductLog[-Exp[-x^2-1]] + 1)/(2 x^2), {x, 0, 2n}, Assumptions -> x>0], {n, 0, 15}] (* Vladimir Reshetnikov, Nov 09 2015 *)
Numerator@Table[KroneckerDelta[n] + 2^n (3n+2)! Sum[((-1)^j 2^i StirlingS2[2n+i+j+1, j])/((2n+i+j+1)! (2n-i+1)! (i-j)! (n+i+1)), {i, 1, 2n+1}, {j, 1, i}], {n, 0, 15}] (* Vladimir Reshetnikov, Nov 09 2015 *)
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PROG
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(PARI) {a(n) = my(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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KEYWORD
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sign,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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