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 A145176 Numerators of coefficients in series expansion of 1/(Bernoulli trial entropy). 3

%I

%S 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,

%T 853,551,173,107,1,7,1,1,19,13579,1313,307,203,7,1,1,1,1679,251,1081,

%U 5969,1681,1169,5,3,1,1,1537,3169,4913,13583,3481,7819,101,11,5,1,1,18167

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

%C 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,

%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 See A145177 for the denominators of R[n,k] and A145178 for numerators scaled to denominators given by A091137.

%H Robert Israel, <a href="/A145176/b145176.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 141 do

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

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

%p 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. A145177, A145178, A091137.

%K frac,nonn,tabl

%O 1,12

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

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Last modified June 25 09:48 EDT 2019. Contains 324347 sequences. (Running on oeis4.)