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A307974
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Inverse binomial transform of the "original" Bernoulli numbers [A164555(n)/A027642(n)] with 1 and 1/2 swapped. Numerators.
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1
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1, 1, -4, 2, -38, 3, -73, 4, -68, 5, -179, 6, -9218, 7, -19, 8, -3976, 9, 18143, 10, -89038, 11, 426463, 12, -118199108, 13, 4276511, 14, -11874736822, 15, 4307920527007, 16, -3854660524816, 17, 1288843929131, 18, -13157635776544491194, 19, 1464996956920721, 20, -130541359248224699708
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OFFSET
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0,3
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COMMENTS
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Denominators: 2, 2, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, ... .
Denominators 3, 15, 21, 15, 33, 1365, 3, 255, ... coincide with cosecant numbers A001897, except 1 (conjectured).
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LINKS
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FORMULA
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a(2*n+1) = n+1 (conjectured).
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EXAMPLE
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Successive differences show the data in the first column:
1/2, 1, 1/6, 0, -1/30, 0, 1/42, 0, ...
1/2, -5/6, -1/6, -1/30, 1/30, 1/42, ...
-4/3, 2/3, 2/15, 1/15, -1/105, ...
2, -8/15, -1/15, -8/105, ...
-38/15, 7/15, -1/105, ...
3, -10/21, ...
-73/21, ...
... .
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MATHEMATICA
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m = 40;
b[n_] = BernoulliB[n]; b[0] = 1/2; b[1] = 1;
a[n_] := Sum[(-1)^(n - k)*Binomial[n, k]*b[k], {k, 0, m}] // Numerator;
Table[a[n], {n, 0, m}]
(* Second program: *)
m = 40;
bb = CoefficientList[Series[x/(1 - Exp[-x]), {x, 0, m}], x]*Range[0, m]!;
bb[[1]] = 1/2; bb[[2]] = 1;
a[n_] := Differences[bb, n][[1]] // Numerator;
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CROSSREFS
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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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