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%I #10 Sep 08 2022 08:46:05
%S 1,12,178,2688,35995,407992,3952943,33615105,257526804,1815863659,
%T 11982128854,74936243346,448516091145,2588488505682,14488775962673,
%U 79019272700951,421449109356322,2204733396548381,11340131754176896,57464577063754608
%N The hyper-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).
%F a(n) = -2026 -14671*n/12 -8077*n^2/24 -605*n^3/12 -83*n^4/24 +2^n*(2961/2 -9783*n/20 -1049*n^2/8 -147*n^3/8 +9*n^4/8 +21*n^5/40) +4^n*(1093/2-279*n/4 +51*n^2/2 -27*n^3/2 +9*n^4/4).
%F G.f.: (1 -25*x +364*x^2 -2888*x^3 +16604*x^4 -77320*x^5 +259299*x^6 -567034*x^7 +849760*x^8 -1145072*x^9 +1576816*x^10 -1535840*x^11 +730496*x^12 -160768*x^13 +83968*x^14 -8192*x^15) / ((1-x)^5*(1-2*x)^6*(1-4*x)^5).
%p a := proc (n) options operator, arrow: -2026-(14671/12)*n-(8077/24)*n^2-(605/12)*n^3-(83/24)*n^4+2^n*(2961/2-(9783/20)*n-(1049/8)*n^2-(147/8)*n^3+(9/8)*n^4+(21/40)*n^5)+4^n*(1093/2-(279/4)*n+(51/2)*n^2-(27/2)*n^3+(9/4)*n^4) end proc: seq(aa(n), n = 0 .. 25);
%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m);Coefficients(R!((1 -25*x +364*x^2 -2888*x^3 +16604*x^4 -77320*x^5 +259299*x^6 -567034*x^7 +849760*x^8 -1145072*x^9 +1576816*x^10 -1535840*x^11 +730496*x^12 -160768*x^13 +83968*x^14 -8192*x^15) / ((1-x)^5*(1-2*x)^6*(1-4*x)^5))); // _Bruno Berselli_, Aug 08 2013
%Y Cf. A227713.
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Aug 07 2013
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