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A145176 Numerators of coefficients in series expansion of 1/(Bernoulli trial entropy). 3
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; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

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*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

See A145177 for the denominators of R[n,k] and A145178 for numerators scaled to denominators given by 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 141 do

  C:= coeff(S, x, n);

  for k from 1 to n do T[n, k]:= numer(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. A145177, A145178, A091137.

Sequence in context: A021855 A256846 A241837 * A093205 A156536 A110191

Adjacent sequences:  A145173 A145174 A145175 * A145177 A145178 A145179

KEYWORD

frac,nonn,tabl

AUTHOR

Tilman Neumann, Oct 03 2008, Oct 04 2008

STATUS

approved

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Last modified June 27 02:59 EDT 2017. Contains 288777 sequences.