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

Author

Tilman Neumann, Oct 03 2008, Oct 04 2008

Keywords

• frac
• nonn
• tabl

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).

Crossrefs

Cf. A145177, A145178, A091137.

Programs

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

MuPAD

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