OFFSET
0,3
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 median child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read paper).
Sum of terms in row n = A002212(n+1).
Column 0 yields the aerated Catalan numbers (1,0,1,0,2,0,5,0,14,...).
T(n,n) = 3^n (see A000244).
Sum_{k=0..n} k*T(n,k) = 3*A026376(n) (n>=1).
LINKS
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
FORMULA
T(n,k) = [3^k/(n+1)]binomial(n+1,k)*binomial(n+1-k,(n-k)/2) (0<=k<=n).
G.f.: G=G(t,z) satisfies G=1+3tzG+z^2*G^2.
EXAMPLE
Triangle starts:
1;
0, 3;
1, 0, 9;
0, 9, 0, 27;
2, 0, 54, 0, 81;
MAPLE
T:=proc(n, k) if n-k mod 2 = 0 then 3^k*binomial(n+1, k)*binomial(n+1-k, (n-k)/2)/(n+1) else 0 fi end: for n from 0 to 11 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Dec 19 2006
STATUS
approved