OFFSET
2,2
COMMENTS
For n > 2 the n-th row has n-2 terms.
The difference between this sequence and A091320 is that this sequence considers the degrees of all vertices whereas A091320 ignores the degree of the root vertex. - Andrew Howroyd, Nov 06 2017
LINKS
E. Deutsch and M. Noy, Statistics on non-crossing trees, Discrete Math., 254 (2002), 75-87.
FORMULA
T(n, k) = U(n, k-1) - U(n, k) + binomial(n-1, k)*Sum_{j=0..k-1} binomial(n-1, j)*binomial(n-k-1, k-1-j)*2^(n-2*k+j)/(n-1) where U(n,k) = 2*binomial(n-2, k)*Sum_{j=0..k-1} binomial(n-1, j)*binomial(n-k-2, k-1-j)*2^(n-1-2*k+j)/(n-2) for 2 < k < n. - Andrew Howroyd, Nov 06 2017
EXAMPLE
Triangle begins:
1;
3;
8, 4;
20, 30, 5;
48, 144, 75, 6;
...
MATHEMATICA
U[n_, k_] := 2 Binomial[n - 2, k] Sum[Binomial[n - 1, j] Binomial[n - k - 2, k - 1 - j] 2^(n - 1 - 2k + j), {j, 0, k - 1}]/(n - 2);
W[n_, k_] := Binomial[n - 1, k] Sum[Binomial[n - 1, j] Binomial[n - k - 1, k - 1 - j] 2^(n - 2k + j), {j, 0, k - 1}]/(n - 1);
T[n_, k_] := If[n < 3, n == 2 && k == 2, If[1 < k && k < n, U[n, k - 1] - U[n, k] + W[n, k]]];
Table[T[n, k] /. True -> 1, {n, 2, 10}, {k, 2, n-Boole[n>2]}] // Flatten (* Jean-François Alcover, Sep 06 2019, from PARI *)
PROG
(PARI)
U(n, k) = 2*binomial(n-2, k)*sum(j=0, k-1, binomial(n-1, j)*binomial(n-k-2, k-1-j)*2^(n-1-2*k+j))/(n-2);
W(n, k) = binomial(n-1, k)*sum(j=0, k-1, binomial(n-1, j)*binomial(n-k-1, k-1-j)*2^(n-2*k+j))/(n-1);
T(n, k) = if(n<3, n==2&&k==2, if(1<k&&k<n, U(n, k-1)-U(n, k)+W(n, k)));
for(n=2, 10, for(k=2, n-(n>2), print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 06 2017
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jul 06 2002
EXTENSIONS
Offset corrected by Andrew Howroyd, Nov 06 2017
More terms from Sean A. Irvine, Sep 12 2024
STATUS
approved