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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..28.

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, i<j)+(if n=2k, (a(k)-1)a(k)/2), n>1. a(1)=2.

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-Fran├žois Alcover, Feb 20 2012, from formula *)

PROG

(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))

CROSSREFS

Cf. A063895.

Sequence in context: A029728 A135547 A146307 * A268619 A024500 A318870

Adjacent sequences:  A063891 A063892 A063893 * A063895 A063896 A063897

KEYWORD

easy,nonn,nice

AUTHOR

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

STATUS

approved

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Last modified May 14 07:13 EDT 2021. Contains 343879 sequences. (Running on oeis4.)