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%I #11 Mar 30 2015 13:10:44
%S 1,-1,4,-5,6,-10,22,-29,40,-66,100,-146,222,-344,534,-797,1208,-1846,
%T 2794,-4230,6430,-9780,14836,-22514,34206,-51936,78826,-119684,181744,
%U -275940,418966,-636125,965848,-1466438,2226482,-3380510,5132678
%N L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} x^(2^n-1) ).
%C Limit a(n+1)/a(n) = -1.518310626574179412829374208878425316378446155444786132846991305715550168878405863954219813056513...
%H Robert Israel, <a href="/A162415/b162415.txt">Table of n, a(n) for n = 1..4962</a>
%F a(n) = (n+1)*sum(m=1..n+1, T(n+1,m)/m*(-1)^(m+1)), where T(n,m):=(1+(-1)^(n-m))/2*sum(k=1..(n-m)/2, binomial(m,k)*T((n-m)/2,k)), T(n,n)=1. - _Vladimir Kruchinin_, Mar 18 2015
%e G.f.: L(x) = x - x^2/2 + 4*x^3/3 - 5*x^4/4 + 6*x^5/5 - 10*x^6/6 +-...
%e where L(x) = log(1 + x + x^3 + x^7 + x^15 + x^31 +...+ x^(2^n-1) +...).
%p N:= 100: # to get a(1) to a(N)
%p L:= ln(add(x^(2^j-1), j= 0 .. ceil(log[2](N)))):
%p S:= series(L,x,N+1):
%p seq(coeff(S,x,n)*n, n=1..N); # _Robert Israel_, Mar 18 2015
%o (PARI) {a(n)=local(L=log(sum(m=0,#binary(n),x^(2^m-1))+x*O(x^n)));n*polcoeff(L,n)}
%o (Maxima)
%o T(n,m):=if n=m then 1 else (1+(-1)^(n-m))/2*sum(binomial(m,k)*T((n-m)/2,k),k,1,(n-m)/2); makelist(n*sum(T(n,m)/m*(-1)^(m+1),m,1,n),n,1,20); /* _Vladimir Kruchinin_, Mar 18 2015 */
%Y Cf. A162416.
%K sign
%O 1,3
%A _Paul D. Hanna_, Jul 02 2009
%E Offset corrected by _Robert Israel_, Mar 18 2015
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