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A122253
Binet's factorial series. Denominators of the coefficients of a convergent series for the logarithm of the Gamma function.
1
12, 12, 360, 60, 280, 168, 5040, 180, 11880, 264, 240240, 10920, 13104, 720, 367200, 3060, 813960, 15960, 1053360, 27720, 3825360, 16560, 5023200, 163800, 982800, 3024, 2630880, 6960, 33227040, 229152, 116867520, 235620, 282744, 2520, 1612119600, 7676760, 46060560
OFFSET
1,1
COMMENTS
See A122252 for references and formulas.
LINKS
Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arXiv preprint arXiv:1602.00336 [math.NT], 2016.
FORMULA
c(n) = (1/n)*Integral_{x=0..1} x^n*(x - 1/2) dx.
EXAMPLE
c(1) = (1/1)*Integral_{x=0..1} x*(x - 1/2) dx = Integral_{x=0..1} (x^2 - x/2) dx = (x^3/3 - x^2/4)|{x=1} - (x^3/3 - x^2/4)|{x=0} = 1/12.
MAPLE
r := n -> add((-1)^(n-j)*Stirling1(n, j)*j/((j+1)*(j+2)), j = 1..n)/(2*n):
a := n -> denom(r(n)); seq(a(n), n = 1..37); # Peter Luschny, Sep 22 2021
MATHEMATICA
Rising[z_, n_Integer/; n>0] := z Rising[z + 1, n - 1]; Rising[z_, 0] := 1; c[n_Integer/; n>0] := Integrate[Rising[x, n] (x - 1/2), {x, 0, 1}] / n;
PROG
(PARI) a(n) = denominator(sum(j=1, n, (-1)^(n-j)*stirling(n, j, 1)*j/((j+1)*(j+2)))/(2*n)); \\ Michel Marcus, Sep 22 2021
CROSSREFS
Cf. A122252 (numerators), A001163, A001164.
Sequence in context: A038338 A221796 A222298 * A156456 A077180 A105745
KEYWORD
easy,frac,nonn
AUTHOR
Paul Drees (zemyla(AT)gmail.com), Aug 27 2006
STATUS
approved