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%I #4 Mar 30 2012 18:37:26
%S 1,3,8,13,23,32,49,59,80,93,127,144,185,203,256,269,319,328,401,419,
%T 504,525,639,656,761,763,904,917,1063,1064,1241,1227,1368,1317,1503,
%U 1480,1681,1659,1928,1909,2143,2080,2393,2371,2696,2653,3055,2992,3305,3147
%N G.f.: A(x) = exp( Sum_{n>=1} A163659(2n)*x^n/n ), where x*exp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).
%C Compare with g.f. of A171238: exp( Sum_{n>=1} A163659(3n)*x^n/n ).
%F G.f. satisfies: A(x) = A(x^2)*(1+x)*(1-x^3)^2/[(1-x)^2*(1+x^3)].
%e G.f.: A(x) = 1 + 3*x + 8*x^2 + 13*x^3 + 23*x^4 + 32*x^5 + 49*x^6 +...
%e log(A(x)) = 3*x + 7*x^2/2 - 6*x^3/3 + 15*x^4/4 + 3*x^5/5 - 14*x^6/6 + 3*x^7/7 + 31*x^8/8 - 6*x^9/9 +...+ A163659(2n)*x^n/n +...
%o (PARI) {A002487(n)=local(c=1, b=0); while(n>0, if(bitand(n, 1), b+=c, c+=b); n>>=1); b}
%o {A163659(n)=n*polcoeff(log(sum(k=0, n, A002487(k+1)*x^k)+x*O(x^n)), n)}
%o {a(n)=polcoeff(exp(sum(k=1, n, A163659(2*k)*x^k/k)+x*O(x^n)), n)}
%Y Cf. A163658, A163659, A171238, A002487.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 07 2011
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