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A153240 Balance of general trees as ordered by A014486, variant A.

%I A153240 #4 Mar 31 2012 13:21:17
%S A153240 0,0,0,0,0,1,-1,0,0,0,1,0,2,2,-1,0,-2,0,1,-2,-1,0,0,0,1,1,2,2,-1,1,0,
%T A153240 3,3,0,3,3,3,-1,0,-1,1,1,-2,-1,-3,0,1,-3,0,2,2,-2,-1,-3,-1,0,-3,-2,0,
%U A153240 1,-3,-2,-1,0,0,0,1,1,2,2,0,2,2,3,3,2,3,3,3,-1,0,0,2,2,-2,1,0,4,4,1,4
%N A153240 Balance of general trees as ordered by A014486, variant A.
%C A153240 This differs from variant A153241 only in that if the degree of the tree is odd (i.e. A057515(n) = 1 mod 2), then the balance of the center-subtree is always taken into account.
%C A153240 Note that for all n, Sum_{i=A014137(n)}^A014138(n) a(i) = 0.
%H A153240 A. Karttunen, <a href="/A153240/b153240.txt">Table of n, a(n) for n = 0..2055</a>
%e A153240 A014486(25) encodes the following general tree:
%e A153240 ......o
%e A153240 ......|
%e A153240 o.o...o.o
%e A153240 .\.\././
%e A153240 ....*..
%e A153240 which consists of four subtrees, of which the second from right is one larger than the others, so we have a(25) = (0+1)-(0+0) = 1.
%o A153240 (MIT Scheme:)
%o A153240 (define (A153240 n) (gentree-deep-balance (A014486->parenthesization (A014486 n))))
%o A153240 (define (gentree-deep-balance l) (let ((r (reverse l))) (let loop ((i 0) (j (- (length l) 1)) (l l) (r r) (z 0)) (cond ((= i j) (+ z (gentree-deep-balance (car l)))) ((> i j) z) (else (loop (+ i 1) (- j 1) (cdr l) (cdr r) (+ z (- (count-pars (car r)) (count-pars (car l))))))))))
%o A153240 (define (count-pars a) (cond ((not (pair? a)) 0) (else (+ 1 (count-pars (car a)) (count-pars (cdr a))))))
%Y A153240 Differs from variant A153241 for the first time at n=268, where A153241(268) = 1, while a(268)=2. Note that (A014486->parenthesization (A014486 268)) = (() (() (())) (())). a(A061856(n)) = 0 for all n. Cf. also A153239.
%K A153240 sign
%O A153240 0,13
%A A153240 _Antti Karttunen_, Dec 21 2008