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A145177 Denominators of rational coefficients in series expansion of 1/(Bernoulli trial entropy). 3

%I

%S 2,6,4,12,6,8,20,9,8,16,30,90,48,12,32,42,720,2160,12,96,64,56,2520,

%T 1440,540,576,32,128,72,25200,10080,2592,1728,24,384,256,90,700,

%U 302400,22680,5184,4320,256,96,512,110,75600,6720,21600,108864,34560,34560,288

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

%C 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,

%C where S(x) = -x*log(x) - (1-x)*log(1-x) is the Bernoulli trial entropy.

%C The series is

%C 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)

%C = 1/(x*(1-log(x))) * (1 + sum_{n=1..inf} x^n * sum_{k=1..n} R[n,k]/(1-log(x))^k).

%C The first rationals R[n,k] are

%C 1/2

%C 1/6 1/4

%C 1/12 1/6 1/8

%C 1/20 1/9 1/8 1/16

%C 1/30 7/90 5/48 1/12 1/32

%C 1/42 41/720 181/2160 1/12 5/96 1/64

%C 1/56 109/2520 97/1440 41/540 35/576 1/32 1/128

%C The LCM of the rows of T[n,k], i.e., A003418(A145177(n,1), ..., A145177(n,n)), is just A091137(n).

%C See A145176 for the numerators of R[n,k] and A145178 for the numerators scaled to denominators A091137.

%H Robert Israel, <a href="/A145177/b145177.txt">Table of n, a(n) for n = 1..10011</a>(first 141 rows, flattened)

%p f:= -x*log(x)-(1-x)*log(1-x):

%p S:= map(normal,eval(series(x*(1-ln(x))/f, x, 12),ln(x)=1-1/t)):

%p for n from 1 to 10 do

%p C:= coeff(S,x,n);

%p for k from 1 to n do T[n,k]:= denom(coeff(C,t,k)) od

%p od:

%p seq(seq(T[n,k],k=1..n),n=1..10); # _Robert Israel_, Jul 09 2015

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

%Y Cf. A003418, A091137, A145176, A145178.

%K frac,nonn,tabl

%O 1,1

%A _Tilman Neumann_, Oct 03 2008, Oct 04 2008

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Last modified April 26 11:11 EDT 2017. Contains 285444 sequences.