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Search: id:A145176
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| A145176 |
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Numerators of coefficients in series expansion of 1/(Bernoulli trial entropy) |
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+0
3
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| 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 1, 1, 1, 41, 181, 1, 5, 1, 1, 109, 97, 41, 35, 1, 1, 1, 853, 551, 173, 107, 1, 7, 1, 1, 19, 13579, 1313, 307, 203, 7, 1, 1, 1, 1679, 251, 1081, 5969, 1681, 1169, 5, 3, 1, 1, 1537, 3169, 4913, 13583, 3481, 7819, 101, 11, 5, 1, 1, 18167
(list; table; graph; listen)
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OFFSET
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1,12
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COMMENT
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This triangle T[n,k] is given by the numerators 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
See A145177 for the denominators of R[n,k] and A145178 for numerators scaled to denominators given by 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. A145177, A145178, A091137
Sequence in context: A144923 A096435 A021855 this_sequence A093205 A156536 A110191
Adjacent sequences: A145173 A145174 A145175 this_sequence A145177 A145178 A145179
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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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