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A302120 Absolute value of the numerators of a series converging to Euler's constant. 2
3, 11, 1, 311, 5, 7291, 243, 14462317, 3364621, 3337014731, 3155743303, 65528247068741, 2627553901, 1439156737843967, 2213381206625, 21757704362231905789, 2627003970197650333, 64925181492079668050329, 523317843775891637, 161371847993975070290712761, 78461950306245817433389909 (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

G. C. Greubel, Table of n, a(n) for n = 1..250

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) = abs(Numerators 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

Numerators of 3/4, -11/96, -1/72, -311/46080, -5/1152, -7291/2322432, ...

MAPLE

a:= proc(n) abs(numer((1/2)*(-1)^(n+1)*(add(Stirling1(n-1, l)*((-1/2)^(l+1)+1)/(l+1), l = 0 .. n-1))/(n)!+(-1)^(n+1)*(add(Stirling1(n, l)/(l+1), l = 1 .. n))/(n*(n)!))) end proc: seq(a(n), n=1..23);

MATHEMATICA

a[n_] := Numerator[(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[Abs[a[n]], {n, 1, 24}]

PROG

(PARI) a(n) = abs(numerator((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!)))

(MAGMA) [3] cat [Abs(Numerator( (1/2)*(-1)^(n+1)*(&+[StirlingFirst(n-1, k)*((-1/2)^(k+1) + 1)/(k+1): k in [1..n-1]])/Factorial(n) + (-1)^(n+1)*(&+[StirlingFirst(n, k)/(k+1): k in [1..n]])/(n*Factorial(n)) )): n in [2..30]]; // G. C. Greubel, Oct 29 2018

CROSSREFS

Cf. A302121 (denominators of this series), A262856, A262858.

Sequence in context: A069604 A098332 A096663 * A133369 A110123 A110221

Adjacent sequences:  A302117 A302118 A302119 * A302121 A302122 A302123

KEYWORD

frac,nonn

AUTHOR

Iaroslav V. Blagouchine, Apr 01 2018

STATUS

approved

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Last modified December 9 22:37 EST 2018. Contains 318032 sequences. (Running on oeis4.)