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

%I A145176 #12 Jan 16 2025 10:49:23
%S A145176 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 A145176 853,551,173,107,1,7,1,1,19,13579,1313,307,203,7,1,1,1,1679,251,1081,
%U A145176 5969,1681,1169,5,3,1,1,1537,3169,4913,13583,3481,7819,101,11,5,1,1,18167
%N A145176 Numerators of coefficients in series expansion of 1/(Bernoulli trial entropy).
%C A145176 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 A145176 where S(x) = - x*log(x) - (1-x)*log(1-x) is the Bernoulli trial entropy.
%C A145176 The series is
%C A145176 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 A145176 = 1/(x*(1-log(x))) * (1 + sum_{n=1..inf} x^n * sum_{k=1..n} R[n,k]/(1-log(x))^k)
%C A145176 The first rationals R[n,k] are
%C A145176 1/2
%C A145176 1/6 1/4
%C A145176 1/12 1/6 1/8
%C A145176 1/20 1/9 1/8 1/16
%C A145176 1/30 7/90 5/48 1/12 1/32
%C A145176 1/42 41/720 181/2160 1/12 5/96 1/64
%C A145176 1/56 109/2520 97/1440 41/540 35/576 1/32 1/128
%C A145176 See A145177 for the denominators of R[n,k] and A145178 for numerators scaled to denominators given by A091137.
%H A145176 Robert Israel, <a href="/A145176/b145176.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows, flattened).
%p A145176 f:= -x*log(x)-(1-x)*log(1-x):
%p A145176 S:= map(normal,eval(series(x*(1-ln(x))/f, x, 12),ln(x)=1-1/t)):
%p A145176 for n from 1 to 141 do
%p A145176   C:= coeff(S,x,n);
%p A145176   for k from 1 to n do T[n,k]:= numer(coeff(C,t,k));
%p A145176  od
%p A145176 od:
%p A145176 seq(seq(T[n,k],k=1..n),n=1..10); # _Robert Israel_, Jul 09 2015
%o A145176 (MuPAD) ORDER:=14: expand(_invert(series(-x*ln(x)-(1-x)*ln(1-x), x=0)));
%Y A145176 Cf. A145177, A145178, A091137.
%K A145176 frac,nonn,tabl
%O A145176 1,12
%A A145176 _Tilman Neumann_, Oct 03 2008, Oct 04 2008