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A145177 Denominators of rational coefficients in series expansion of 1/(Bernoulli trial entropy). 3
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; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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 x=0,

where S(x) = -x*log(x) - (1-x)*log(1-x) is the Bernoulli trial entropy.

The series is

1/S(x) = 1/(x*(1-log(x))) + sum_{n=1..inf} x^(n-1) * sum_{k=1..n} R[n,k]/(1-log(x))^(k+1)

= 1/(x*(1-log(x))) * (1 + sum_{n=1..inf} x^n * sum_{k=1..n} R[n,k]/(1-log(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.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10011(first 141 rows, flattened)

MAPLE

f:= -x*log(x)-(1-x)*log(1-x):

S:= map(normal, eval(series(x*(1-ln(x))/f, x, 12), ln(x)=1-1/t)):

for n from 1 to 10 do

  C:= coeff(S, x, n);

  for k from 1 to n do T[n, k]:= denom(coeff(C, t, k)) od

od:

seq(seq(T[n, k], k=1..n), n=1..10); # Robert Israel, Jul 09 2015

PROG

(Other) ORDER:=14: expand(_invert(series(-x*ln(x)-(1-x)*ln(1-x), x=0)));

CROSSREFS

Cf. A003418, A091137, A145176, A145178.

Sequence in context: A266013 A222423 A059909 * A007517 A072946 A134000

Adjacent sequences:  A145174 A145175 A145176 * A145178 A145179 A145180

KEYWORD

frac,nonn,tabl

AUTHOR

Tilman Neumann, Oct 03 2008, Oct 04 2008

STATUS

approved

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Last modified May 26 10:37 EDT 2017. Contains 287095 sequences.