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A264148
Numerators of rational coefficients related to Stirling's asymptotic series for the Gamma function.
5
1, 2, 1, -4, 1, 8, -139, 16, -571, -8992, 163879, -334144, 5246819, 698752, -534703531, 23349012224, -4483131259, -1357305243136, 432261921612371, -6319924923392, 6232523202521089, 8773495082018816, -25834629665134204969, 49004477022654464, -1579029138854919086429
OFFSET
0,2
COMMENTS
The rational numbers SGGS = A264148/A264149 (SGGS stands for 'Stirling Generalized Gamma Series') are a supersequence of the coefficients in Stirling's asymptotic series for the Gamma function A001163/A001164 and of an asymptotic expansion of Ramanujan A090804/A065973, further they appear in scaled form in an expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function A005447/A005446.
Ramanujan's asymptotic expansion theta(n) = 1/3+4/(135n)-8/(2835n^2)- ... is considered in the literature also in the form 1-theta(n) (see for example formula (5) in the Choi link). It is this form to which we refer here.
LINKS
K. P. Choi, On the medians of gamma distributions and an equation of Ramanujan, Proceedings of the American Mathematical Society 121:1 (May, 1994), pp. 245-251. [From Vladimir Reshetnikov]
G. Nemes, On the coefficients of the asymptotic expansion of n!, J. Integer Seqs. 13 (2010), 5. [From Vladimir Reshetnikov]
FORMULA
Let SGGS(n) = h(n)*doublefactorial(n-1) where h(n) = 1 for n<=0 and for n>0 defined by the recurrence (h(k-1)/k - Sum_{j=1..k-1}((h(k-j)*h(j))/(j+1))/ (1+1/(k+1))) then a(n) = numerator(SGGS(n)).
MAPLE
h := proc(k) option remember; local j; `if`(k<=0, 1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
SGGS := n -> h(n)*doublefactorial(n-1):
A264148 := n -> numer(SGGS(n)): seq(A264148(n), n=0..24);
MATHEMATICA
h[k_] := h[k] = If[k <= 0, 1, (h[k - 1]/k - Sum[h[k - j]*h[j]/(j + 1), {j, 1, k - 1}]) / (1 + 1/(k + 1))]; a[n_] := h[n]* Factorial2[n - 1] // Numerator; Table[a[n], {n, 0, 24}]
PROG
(Sage)
def A264148(n):
@cached_function
def h(k):
if k<=0: return 1
S = sum((h(k-j)*h(j))/(j+1) for j in (1..k-1))
return (h(k-1)/k-S)/(1+1/(k+1))
return numerator(h(n)*(n-1).multifactorial(2))
print([A264148(n) for n in (0..17)])
CROSSREFS
A264148(n) = numerator(SGGS(n)).
A264149(n) = denominator(SGGS(n)).
A001163(n) = numerator(SGGS(2*n)) = numerator(SGGS(2*n)/2^(n+1)).
A001164(n) = denominator(SGGS(2*n)).
A090804(n) = numerator(SGGS(2*n+1)).
A065973(n) = denominator(SGGS(2*n+1)) = denominator(SGGS(2*n+1)/2^(n+1)).
A005447(n+1) = numerator(SGGS(n)/2^(n+1)).
A264150(n) = numerator(SGGS(2*n+1)/2^(n+1)).
Sequence in context: A112481 A134851 A355227 * A038001 A147080 A146418
KEYWORD
sign,frac
AUTHOR
Peter Luschny, Nov 05 2015
STATUS
approved