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A277000
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Numerators of an asymptotic series for the Gamma function (even power series).
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5
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1, -1, 19, -2561, 874831, -319094777, 47095708213409, -751163826506551, 281559662236405100437, -49061598325832137241324057, 5012066724315488368700829665081, -26602063280041700132088988446735433, 40762630349420684160007591156102493590477
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OFFSET
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0,3
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COMMENTS
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Let y = x+1/2 then Gamma(x+1) ~ sqrt(2*Pi)*((y/E)*Sum_{k>=0} r(k)/y^(2*k))^y as x -> oo and r(k) = A277000(k)/A277001(k) (see example 6.1 in the Wang reference).
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LINKS
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FORMULA
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a(n) = numerator(b(2*n)) with b(n) = Y_{n}(0, z_2, z_3,..., z_n)/n! with z_k = k!*Bernoulli(k,1/2)/(k*(k-1)) and Y_{n} the complete Bell polynomials.
The rational numbers have the recurrence r(n) = (1/(2*n))*Sum_{m=0..n-1} Bernoulli(2*m+2,1/2)*r(n-m-1)/(2*m+1)) for n>=1, r(0)=1. - Peter Luschny, Sep 30 2016
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EXAMPLE
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The underlying rational sequence starts:
1, 0, -1/24, 0, 19/5760, 0, -2561/2903040, 0, 874831/1393459200, 0, ...
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MAPLE
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b := n -> CompleteBellB(n, 0, seq((k-2)!*bernoulli(k, 1/2), k=2..n))/n!:
# Alternatively the rational sequence by recurrence:
R := proc(n) option remember; local k; `if`(n=0, 1,
add(bernoulli(2*m+2, 1/2)* R(n-m-1)/(2*m+1), m=0..n-1)/(2*n)) end:
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MATHEMATICA
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CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
b[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1/2], {k, 2, n}]]]/n!;
a[n_] := Numerator[b[2n]];
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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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STATUS
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approved
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