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%I #14 Jan 20 2020 21:27:21
%S 1,2,4,10,30,110,494,2734,18734,159278,1685550,22268974,367653934,
%T 7597868078,196929315886,6402998805550,261393582040110,
%U 13416320169124910,865576139256079406,70227589169019724846,7172766017169503134766,921829147482582383174702
%N O.g.f.: A(x) = Sum_{n>=0} x^n*Product_{k=0..n} (1 + 2^k*x).
%F G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (1+x*2^k)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2013
%F a(n) = Sum_{k=0..floor((n+1)/2)} q-binomial(n-k+1,k)*2^binomial(k,2), where q-binomial is triangle A022166, that is, with q=2. - _Vladimir Kruchinin_, Jan 21 2020
%e A(x) = (1+x) + x*(1+x)*(1+2x) + x^2*(1+x)*(1+2x)*(1+4x) + x^3*(1+x)*(1+2x)*(1+4x)*(1+8x) +...
%o (PARI) a(n)=polcoeff(sum(k=0,n,x^k*prod(j=0,k,1+2^j*x+x*O(x^n))),n)
%Y Cf. A022166.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 28 2006
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