A302120 Absolute value of the numerators of a series converging to Euler's constant.
3, 11, 1, 311, 5, 7291, 243, 14462317, 3364621, 3337014731, 3155743303, 65528247068741, 2627553901, 1439156737843967, 2213381206625, 21757704362231905789, 2627003970197650333, 64925181492079668050329, 523317843775891637, 161371847993975070290712761, 78461950306245817433389909
Offset: 1
Examples
Numerators of 3/4, -11/96, -1/72, -311/46080, -5/1152, -7291/2322432, ...
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.
Programs
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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
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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);
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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}] -
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!)))
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.
Comments