A145176 -id:A145176 - cp's OEIS frontend

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

Tilman Neumann, Oct 03 2008, Oct 04 2008

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.

Robert Israel, Table of n, a(n) for n = 1..10011(first 141 rows, flattened)

Cf. A003418, A091137, A145176, A145178.

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

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

A145178 Numerators of coefficients of series expansion of 1/(Bernoulli trial entropy), scaled to denominators A091137.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 36, 80, 90, 45, 48, 112, 150, 120, 45, 1440, 3444, 5068, 5040, 3150, 945, 2160, 5232, 8148, 9184, 7350, 3780, 945, 50400, 122832, 198360, 242200, 224700, 151200, 66150, 14175, 80640, 196992, 325896, 420160, 429800, 341040, 198450

Offset: 1

Tilman Neumann, Oct 03 2008

These are the numerators A145176 scaled to denominators A091137.

In other words: A145178(n,k)/A091137(n) = A145176(n,k)/A145177(n,k)

Cf. A091137, A145176, A145177

Corrected description of the series. - Tilman Neumann, Oct 04 2008

cp's OEIS Frontend

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

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