OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
Alexander Burstein, Megan Martinez, Pattern classes equinumerous to the class of ternary forests, Permutation Patterns Virtual Workshop, Howard University (2020).
W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, arXiv:math/0504342 [math.CO], 2005.
W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 2.2.
D. S. Hough, Descents in noncrossing trees, Electronic J. Combinatorics 10 (2003), #N13, Theorem 2.2. [Ira M. Gessel, May 10 2010]
FORMULA
T(n, k) = Sum_{i=n..2*n-1} (-1)^(n+k+i)/i*C(i, n)*C(3*n, i+1+n)*C(i-n, k).
T(n,k) = C(n-1+k,n-1)*C(2*n-k,n+1)/n, (0 <= k <= n-1). [Chen et al.] - Emeric Deutsch, Dec 19 2006
O.g.f. equals the series reversion with respect to x of x*(1 + x*(1 - t))/(1 + x)^3. If R(n,t) is the n-th row polynomial of this triangle then R(n, 1+t) is the n-th row polynomial of A089434. - Peter Bala, Jul 15 2012
EXAMPLE
Triangle begins
1;
2, 1;
5, 5, 2;
14, 21, 15, 5;
42, 84, 84, 49, 14;
132, 330, 420, 336, 168, 42;
429, 1287, 1980, 1980, 1350, 594, 132;
1430, 5005, 9009, 10725, 9075, 5445, 2145, 429;
4862, 19448, 40040, 55055, 55055, 40898, 22022, 7865, 1430;
MAPLE
T:=(n, k)->binomial(n-1+k, n-1)*binomial(2*n-k, n+1)/n: for n from 1 to 10 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch, Dec 19 2006
MATHEMATICA
T[n_, k_] := Binomial[n + k - 1, n - 1]*Binomial[2*n - k, n + 1]/n;
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Nov 11 2017, after Emeric Deutsch *)
PROG
(PARI) T(n, k) = binomial(n-1+k, n-1)*binomial(2*n-k, n+1)/n; \\ Andrew Howroyd, Nov 06 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Jun 03 2005
STATUS
approved