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A210244
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Numerators of the polylogarithm li(-n,-1/2)/2.
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3
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-1, -1, 1, 5, -7, -49, -53, 2215, 1259, -14201, -183197, 248885, 9583753, 14525053, -554173253, -4573299625, 99833187251, 215440236599, -1654012631597, -84480933600305, -36267273557287, 10992430255511053, 117548575473066241, -1380910044674479865
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OFFSET
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1,4
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COMMENTS
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Given an integer n>0, consider the infinite series s(n) = li(-n,-1/2)) = SUM((-1)^k)(k^n)/2^k) for k=1,2,... Then s(n)=2*a(n)/A131137(n+1).
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LINKS
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FORMULA
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Recurrence: s(n+1)=(-1/3)*SUM(C(n+1,i)*s(i)), where i=0,1,2,...,n, and C(n,m) are the binomial coefficients, with the starting value of s(0)=2/3.
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EXAMPLE
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s(1)=-2/9, s(2)=-2/27, s(3)=+2/27, s(4)=+10/81.
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MATHEMATICA
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nn = 30; s[0] = 1; Do[s[n+1] = (-1/3) Sum[Binomial[n+1, i] s[i], {i, 0, n}], {n, 0, nn}]; Numerator[Table[s[n], {n, 0, nn}]] (* T. D. Noe, Mar 20 2012 *)
Table[PolyLog[-n, -1/2]/2, {n, 30}] (* T. D. Noe, Mar 23 2012 *)
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PROG
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CROSSREFS
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Denominators: A131137, offset by 1.
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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