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Search: id:A145177
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| A145177 |
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Denominators of rational coefficients in series expansion of 1/(Bernoulli trial entropy) |
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+0
3
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| 2, 6, 4, 12, 6, 8, 20, 9, 8, 16, 30, 90, 48, 12, 32, 42, 720, 2160, 12, 96, 64, 56, 2520, 1440, 540, 576, 32, 128, 72, 25200, 10080, 2592, 1728, 24, 384, 256, 90, 700, 302400, 22680, 5184, 4320, 256, 96, 512, 110, 75600, 6720, 21600, 108864, 34560, 34560, 288
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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This triangle T[n,k] is given by the denominators of rational coefficients R[n,k] appearing in a certain series expansion of 1/S(x) around x0=0,
where S(x) = - x*ln(x) - (1-x)*ln(1-x) is the Bernoulli trial entropy.
The series is
1/S(x) = 1/(x*(1-ln(x))) + sum_{n=1..inf} x^(n-1) * sum_{k=1..n} R[n,k]/(1-ln(x))^(k+1)
= 1/(x*(1-ln(x))) * (1 + sum_{n=1..inf} x^n * sum_{k=1..n} R[n,k]/(1-ln(x))^k)
The first rationals R[n,k] are
1/2
1/6 1/4
1/12 1/6 1/8
1/20 1/9 1/8 1/16
1/30 7/90 5/48 1/12 1/32
1/42 41/720 181/2160 1/12 5/96 1/64
1/56 109/2520 97/1440 41/540 35/576 1/32 1/128
The LCM of the rows of T[n,k], i.e. A003418(A145177(n,1), ..., A145177(n,n)), is just A091137(n).
See A145176 for the numerators of R[n,k] and A145178 for the numerators scaled to denominators A091137.
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PROGRAM
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(Other) ORDER:=14: expand(_invert(series(-x*ln(x)-(1-x)*ln(1-x), x=0)));
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CROSSREFS
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Cf. A003418, A091137, A145176, A145178
Sequence in context: A065880 A090546 A059909 this_sequence A007517 A072946 A134000
Adjacent sequences: A145174 A145175 A145176 this_sequence A145178 A145179 A145180
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KEYWORD
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frac,nonn,tabl
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AUTHOR
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Tilman Neumann (Tilman.Neumann(AT)web.de), Oct 03 2008, Oct 04 2008
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