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A302121 Denominators of a series converging to Euler's constant. 2
4, 96, 72, 46080, 1152, 2322432, 100352, 7431782400, 2090188800, 2452488192000, 2697737011200, 64274810535936000, 2923954176000, 1799694695006208000, 3085190905724928, 33566877054287216640000, 4458100858772520960000, 120538655501945394954240000, 1057781497894797312000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
gamma = 3/4 - 11/96 - 1/72 - 311/46080 - 5/1152 - 7291/2322432 - ..., see formula (104) in the reference below.
LINKS
Ia. V. Blagouchine, Three Notes on Ser's and Hasse's Representations for the Zeta-functions. INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A3, pp. 1-45, 2018. arXiv:1606.02044 [math.NT], 2016.
FORMULA
a(n) = Denominators of ((1/2)*(-1)^(n+1)*(Sum_{l=0..n-1} (S_1(n-1,l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(Sum_{l=1..n} S_1(n,l)/(l+1)))/(n*n!)), where S_1(x,y) are the signed Stirling numbers of the first kind.
EXAMPLE
Denominators of 3/4, -11/96, -1/72, -311/46080, -5/1152, -7291/2322432, ...
MAPLE
a := proc (n) options operator, arrow; denum((1/2)*(-1)^(n+1)*(sum(Stirling1(n-1, l)*((-1/2)^(l+1)+1)/(l+1), l = 0 .. n-1))/factorial(n)+(-1)^(n+1)*(sum(Stirling1(n, l)/(l+1), l = 1 .. n))/(n*factorial(n))) end proc
MATHEMATICA
a[n_] := Denominator[(1/2)*(-1)^(n+1)*(Sum[StirlingS1[n-1, l]*((-1/2)^(l+1) + 1)/(l+1), {l, 0, n-1}])/(n!) + (-1)^(n+1)*(Sum[StirlingS1[n, l]/(l+1), {l, 1, n}])/(n*n!)]; Table[a[n], {n, 1, 24}]
PROG
(PARI) a(n) = denominator((1/2)*(-1)^(n+1)*(sum(l=0, n-1, stirling(n-1, l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(sum(l=1, n, stirling(n, l)/(l+1)))/(n*n!))
CROSSREFS
Cf. A302120 (numerators of this series), A262856, A262858.
Sequence in context: A210857 A220682 A210926 * A188895 A068114 A272109
KEYWORD
frac,nonn
AUTHOR
STATUS
approved

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Last modified June 15 11:51 EDT 2024. Contains 373407 sequences. (Running on oeis4.)