A126187 Sum of the levels of the first leaf (in the preorder traversal) over all hex trees with n edges.

3, 19, 96, 453, 2085, 9513, 43323, 197542, 903141, 4142565, 19067202, 88065360, 408108285, 1897265405, 8846769300, 41368049400, 193950461985, 911564782065, 4294230794520, 20273068467725, 95902496669091, 454528832324919

Offset: 1

Author

Emeric Deutsch, Dec 22 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. A126186.

Programs

Maple

g:=2*(1+3*z-sqrt(1-6*z+5*z^2))/(1-3*z+sqrt(1-6*z+5*z^2))^2: gser:=series(g,z=0,28): seq(coeff(gser,z,n),n = 1..25);

Formula

a(n) = Sum_{k=1..n} k*A126186(n,k).

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

D-finite with recurrence (n-1)*(3*n-1)*(n+4)*a(n) -n*(18*n^2+21*n-19)*a(n-1) +5*n*(3*n+2)*(n-1)*a(n-2)=0. - R. J. Mathar, Jun 17 2016