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%I #9 Apr 30 2014 01:37:32
%S 2,1,2,4,10,25,68,187,532,1532,4492,13308,39870,120433,366656,1123504,
%T 3463050,10729234,33396202,104381385,327477814,1030903514,3255394686,
%U 10309154738,32732315724,104177891232,332308237666,1062192108053
%N Start with x, y; then concatenate each word in turn with all preceding words, getting x y xy xxy yxy xxxy yxxy xyxxy ...; sequence gives number of words of length n. Also binary trees by degree: x y (x,y) (x,(x,y)) (y,(x,y)) (x,(x,(x,y))) (y,(x,(x,y))) ((x,y),(x,(x,y)))...
%F G.f. A(x)=1-sqrt(1-4x+A(x^2)) satisfies A(x)^2-2A(x)+4x-A(x^2)=0, A(0)=0. - _Michael Somos_, Sep 06 2003
%F a(n)=(sum a(i)a(j), i+j=n, i<j)+(if n=2k, (a(k)-1)a(k)/2), n>1. a(1)=2.
%t a[1] = 2; a[n_] := a[n] = Sum[a[k]*a[n-k], {k, 1, Floor[(n-1)/2]}] + If[EvenQ[n], (a[n/2]-1)*a[n/2]/2, 0]; Table[a[n], {n, 1, 28}] (* _Jean-François Alcover_, Feb 20 2012, from formula *)
%o (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=O(x); while(m<=n,m*=2; A=1-sqrt(1-4*x+subst(A,x,x^2))); polcoeff(A,n))
%Y Cf. A063895.
%K easy,nonn,nice
%O 1,1
%A Claude Lenormand (claude.lenormand(AT)free.fr), Aug 29 2001
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