OFFSET
1,3
COMMENTS
Row n contains n terms.
Row sums yield the ternary numbers (A001764).
The average number of nonroot nodes of degree 1 over all noncrossing trees with n edges is 4n(n-1)(2n+1)/(3(3n-1)(3n-2)) ~ 8n/27.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
FORMULA
T(n, k) = [2^k/(n-k)]*binomial(n-1, k)*Sum_{i=1..n-k} (-1)^(n-k-i)*2^(n-k-i)*binomial(n-k, i)*binomial(3i, i-1), 0 <= k < n).
T(n,k) = 2^k*binomial(n-1,k)*A030981(n-k).
EXAMPLE
T(2,0)=1 (/\); T(2,1)=2 (/_, _\ ).
Triangle begins:
1;
1, 2;
4, 4, 4;
11, 24, 12, 8;
41, 88, 96, 32, 16;
MAPLE
T:=proc(n, k) if k<n then 2^k*binomial(n-1, k)*sum((-1)^(n-k-i)*2^(n-k-i)*binomial(n-k, i)*binomial(3*i, i-1), i=1..n-k)/(n-k) else 0 fi end: for n from 1 to 10 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form
MATHEMATICA
T[n_, k_] := If[k<n, 2^k*Binomial[n-1, k]*Sum[(-1)^(n-k-i)*2^(n-k-i)* Binomial[n-k, i]*Binomial[3*i, i-1], {i, 1, n-k}]/(n-k), 0];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 06 2018, from Maple *)
PROG
(PARI) T(n, k)={if(k<n, 2^k*binomial(n-1, k)*sum(i=1, n-k, (-1)^(n-k-i)*2^(n-k-i)*binomial(n-k, i)*binomial(3*i, i-1))/(n-k))}
for(n=1, 10, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 06 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jan 17 2005
STATUS
approved