A145178 -id:A145178 - cp's OEIS frontend

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

Tilman Neumann, Oct 03 2008, Oct 04 2008

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.

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

Cf. A145177, A145178, A091137.

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

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

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

Original entry on oeis.org

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)));

cp's OEIS Frontend

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

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