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

1,1

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