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A182000 G.f.: exp( Sum_{n>=1} 2^A090740(n) * x^n/n ) where A090740(n) = highest exponent of 2 in 3^n-1.

%I A182000 #12 Apr 17 2012 13:03:05
%S A182000 1,2,6,10,22,34,62,90,150,210,326,442,654,866,1230,1594,2198,2802,
%T A182000 3766,4730,6230,7730,9998,12266,15630,18994,23878,28762,35742,42722,
%U A182000 52526,62330,75926,89522,108118,126714,151878,177042,210702,244362,288982,333602,392182
%N A182000 G.f.: exp( Sum_{n>=1} 2^A090740(n) * x^n/n ) where A090740(n) = highest exponent of 2 in 3^n-1.
%H A182000 Paul D. Hanna, <a href="/A182000/b182000.txt">Table of n, a(n) for n = 0..1024</a>
%F A182000 G.f. satisfies: A(x) = (1+x^2)/(1-x)^2 * A(x^2).
%F A182000 Define BISECTIONS: A(x) = B_0(x^2) + x*B_1(x^2), then: B_1(x)/B_0(x) = 2/(1+x).
%e A182000 G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 22*x^4 + 34*x^5 + 62*x^6 +...
%e A182000 The g.f. satisfies:
%e A182000 A(x)/A(x^2) = 1 + 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 +...+ 2*n*x^n +...
%e A182000 The logarithm of the g.f. begins:
%e A182000 log(A(x)) = 2*x + 8*x^2/2 + 2*x^3/3 + 16*x^4/4 + 2*x^5/5 + 8*x^6/6 + 2*x^7/7 + 32*x^8/8 + 2*x^9/9 + 8*x^10/10 + 2*x^11/11 + 16*x^12/12 +...+ 2^A090740(n)*x^n/n +...
%e A182000 where the highest exponents of 2 in 3^n-1, for n>=1, begins:
%e A182000 A090740 = [1,3,1,4,1,3,1,5,1,3,1,4,1,3,1,6,1,3,1,4,1,3,1,5,1,3,1,4,1,...].
%e A182000 The g.f.s of the BISECTIONS begin:
%e A182000 B_0(x) = 1 + 6*x + 22*x^2 + 62*x^3 + 150*x^4 + 326*x^5 + 654*x^6 +...
%e A182000 B_1(x) = 2 + 10*x + 34*x^2 + 90*x^3 + 210*x^4 + 442*x^5 + 866*x^6 +...
%e A182000 where 2 * B_0(x) / B_1(x) = 1+x.
%o A182000 (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,2^valuation(3^m-1,2)*x^m/m)+x*O(x^n)),n)}
%o A182000 for(n=0,40,print1(a(n),", "))
%o A182000 (PARI) {a(n)=local(A=1+x);for(i=1,#binary(n)+1,A=(1+x^2)/(1-x)^2*subst(A,x,x^2+x*O(x^n)));polcoeff(A,n)}
%Y A182000 Cf. A173283, A161809, A182185, A090740.
%K A182000 nonn
%O A182000 0,2
%A A182000 _Paul D. Hanna_, Apr 17 2012