A145176 Numerators of coefficients in series expansion of 1/(Bernoulli trial entropy).

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

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: A256846 A336698 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