A264148 Numerators of rational coefficients related to Stirling's asymptotic series for the Gamma function.

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

G. C. Greubel, Table of n, a(n) for n = 0..360

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

Adjacent sequences: A264145 A264146 A264147 * A264149 A264150 A264151

Keyword

sign,frac

Author

Peter Luschny, Nov 05 2015

Status

approved