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
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
KEYWORD
AUTHOR
Tilman Neumann, Oct 03 2008, Oct 04 2008
STATUS
approved