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%I #18 Sep 08 2022 08:46:05
%S 1,9,90,836,6856,49787,326618,1977322,11244976,60908337,317509874,
%T 1605448440,7920487752,38297112551,182108066522,853884638758,
%U 3956279351760,18143381822941,82466719670866,371917534537524,1665777832832136,7415146800493139
%N The Wiener index of the tree g[n] (n>=0) defined recursively in the following manner: denoting by P[n] the path on n vertices, we define g[0] =P[2] while g[n] (n>=1) is the tree obtained by identifying the roots of 2 copies of g[n-1] and one of the end-vertices of P[n+1]; the root of g[n] is defined to be the other end-vertex of P[n+1].
%C Roughly speaking, g[4], for example, is obtained from the planted full binary tree of height 5 by replacing the edges at the levels 1,2,3,4 with paths of lengths 4, 3, 2, and 1, respectively.
%C The value of a(4) has been checked by the direct evaluation of the Wiener index (using Maple).
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (24,-254,1564,-6225,16820,-31496,40896,-36112,20672,-6912,1024).
%F a(n) = 80 +128*n/3 +19*n^2/2 +5*n^3/6 +2^n*(-64 +38*n +21*n^2/2 +3*n^3/2) +4^n*(-15 -27*n/2 +9*n^2/2).
%F G.f.: (1 -15*x +128*x^2 -602*x^3 +1801*x^4 -3968*x^5 +6016*x^6 -5528*x^7 +3120*x^8 -1344*x^9 +256*x^10) / ((1-x)^4*(1-2*x)^4*(1-4*x)^3).
%e a(1)=9 because g[1] is the tree in the shape of Y and 3*1 + 3*2 = 9.
%p a := proc (n) options operator, arrow: 80+(128/3)*n+(19/2)*n^2+(5/6)*n^3 +2^n*(-64+38*n +(21/2)*n^2+(3/2)*n^3)+4^n*(-15-(27/2)*n+(9/2)*n^2) end proc: seq(a(n), n = 0 .. 25);
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m);Coefficients(R!((1-15*x +128*x^2-602*x^3 +1801*x^4-3968*x^5+6016*x^6 -5528*x^7+3120*x^8 -1344*x^9 +256*x^10)/((1-x)^4*(1-2*x)^4*(1-4*x)^3))); // _Bruno Berselli_, Aug 08 2013
%Y Cf. A227714.
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Aug 07 2013
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