A126323 - OEIS

%I #7 Dec 30 2016 02:31:34

%S 0,3,2,15,80,399,1956,9546,46552,227100,1108698,5417127,26490312,

%T 129645027,634978290,3112277265,15264984260,74919716085,367926876630,

%U 1807912844925,8888531467360,43722603214365,215175747222640

%N Number of branches of length 1 in all hex trees with n edges.

%C A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference).

%H F. Harary and R. C. Read, <a href="https://doi.org/10.1017/S0013091500009135">The enumeration of tree-like polyhexes</a>, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

%F a(n) = Sum_{k=0..n} k*A126321(n,k).

%F G.f.: (1-3z)^2*[2-9z+5z^2-(2-3z)sqrt(1-6z+5z^2)]/[2z^2*sqrt(1-6z+5z^2)].

%F Conjecture: -(n+2)*(1176*n^2+350135*n-2095015)*a(n) +3*(-392*n^3+1110577*n^2-6230546*n-1231580)*a(n-1) +(43512*n^3-9103273*n^2+68264245*n-103131090) *a(n-2) -15*(n-4)*(2744*n^2-398547*n+1917624)*a(n-3)=0. - _R. J. Mathar_, Jun 17 2016

%p g:=(1-3*z)^2*(2-9*z+5*z^2-(2-3*z)*sqrt(1-6*z+5*z^2))/2/z^2/sqrt(1-6*z+5*z^2): gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=0..27);

%Y Cf. A126321.

%K nonn

%O 0,2

%A _Emeric Deutsch_, Dec 25 2006