A063894 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)))...

2, 1, 2, 4, 10, 25, 68, 187, 532, 1532, 4492, 13308, 39870, 120433, 366656, 1123504, 3463050, 10729234, 33396202, 104381385, 327477814, 1030903514, 3255394686, 10309154738, 32732315724, 104177891232, 332308237666, 1062192108053

Offset: 1

Author

Claude Lenormand (claude.lenormand(AT)free.fr), Aug 29 2001

Keywords

• easy
• nonn
• nice

Crossrefs

Cf. A063895.

Programs

Mathematica

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 *)

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))

Formula

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

a(n)=(sum a(i)a(j), i+j=n, i1. a(1)=2.