A126185 Number of nonroot nodes of outdegree 2 in all hex trees with n edges.

0, 0, 0, 3, 29, 198, 1180, 6570, 35196, 184128, 948516, 4835295, 24469005, 123174810, 617662890, 3088403955, 15409111025, 76755126600, 381848749720, 1897825700385, 9425387927295, 46783757341050, 232114595171200

Offset: 0

Author

Emeric Deutsch, Dec 19 2006

Keywords

• nonn

Comments

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).

Links

• F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Crossrefs

Cf. A126183.

Programs

Maple

G:=((2-15*z+30*z^2-15*z^3)*sqrt(1-6*z+5*z^2)-(1-z)^2*(1-5*z)*(2-7*z))/2/z^2/(1-6*z+5*z^2):Gser:=series(G,z=0,31): seq(coeff(Gser,z,n),n=0..26);

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} k*A126183(n,k).

G.f.= [(2-15z+30z^2-15z^3)sqrt(1-6z+5z^2)-(1-5z)(2-7z)(1-z)^2]/[2z^2*(1-6z+5z^2)].

D-finite with recurrence -(n+2)*(n-3)*a(n) +(7*n+1)*(n-2)*a(n-1) -(11*n-15)*(n-2)*a(n-2) +5*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 22 2022