A145177 Denominators of rational coefficients in series expansion of 1/(Bernoulli trial entropy).

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

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 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)

Crossrefs

Cf. A003418, A091137, A145176, A145178.

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 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

MuPAD

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