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A122252 Numerator of the n-th series entry for the convergent form of Stirling's Approximation for the gamma function. log gamma z = (z - 1/2) log z - z + log(2*Pi)/2 + sum(c(n)/(z+1)^(n), {n, 1, infinity}], where z^(n) is the rising factorial. 2
1, 1, 59, 29, 533, 1577, 280361, 69311, 36226519, 7178335, 64766889203, 32128227179, 459253205417, 325788932161, 2311165698322609, 287144996287039, 1215091897184850539, 402833263943353393, 476099430416027805187 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..100

Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arXiv preprint arXiv:1602.00336, 2016

Wikipedia, Stirling's Approximation

FORMULA

c(n) = integral(x^(n)*(x - 1/2), {x, 0, 1}) / n.

EXAMPLE

c(1) = integral(x*(x - 1/2), {x, 0, 1}) / 1 = integral(x^2 - x/2, {x, 0, 1}) = x^3/3 - x^2/4|{x, 0, 1} = 1/12.

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; Numerator@ Array[c, 19] (* updated by Robert G. Wilson v, Aug 15 2015 *)

CROSSREFS

Cf. A001163, A001164, A122253.

Sequence in context: A145532 A152214 A033379 * A119945 A054379 A278372

Adjacent sequences:  A122249 A122250 A122251 * A122253 A122254 A122255

KEYWORD

easy,frac,nonn

AUTHOR

Paul Drees (zemyla(AT)gmail.com), Aug 27 2006

STATUS

approved

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Last modified July 22 20:51 EDT 2019. Contains 325226 sequences. (Running on oeis4.)