OFFSET
0,3
COMMENTS
A Catalan path is a Dyck path of length 2n that ends on the horizontal axis.
LINKS
G. C. Greubel, Rows n = 0..30 of triangle, flattened
FORMULA
Let q-Narayana(n,k) = q-binomial(n, k) * q-binomial(n-1, k) / q-binomial(k+1,1) then the n-th polynomial is Sum_{k=0..n} q-Narayana(n,k).
EXAMPLE
The polynomials start:
1,
1,
2,
3 + q + q^2,
4 + 2q + 4q^2 + 2q^3 + 2q^4,
5 + 3q + 7q^2 + 7q^3 + 8q^4 + 5q^5 + 5q^6 + q^7 + q^8.
The triangle starts:
[n] [row] [sum]
[0] [1] 1
[1] [1] 1
[2] [2] 2
[3] [3, 1, 1] 5
[4] [4, 2, 4, 2, 2] 14
[5] [5, 3, 7, 7, 8, 5, 5, 1, 1] 42
[6] [6, 4, 10, 12, 18, 16, 20, 14, 14, 8, 6, 2, 2] 132
MATHEMATICA
QNarayana[n_, k_]:= QBinomial[n, k, q] QBinomial[n-1, k, q]/QBinomial[k+1, 1, q];
QNarayanaRow[n_]:= Sum[QNarayana[n, k], {k, 0, n}];
Table[CoefficientList[QNarayanaRow[n] // FunctionExpand, q], {n, 0, 8}] // Flatten
PROG
(Sage)
from sage.combinat.q_analogues import q_int, q_binomial
def q_Narayana(n, k, q=None):
if n == 0: return 1
return q_binomial(n, k, q)*q_binomial(n-1, k, q)//q_int(k+1)
def q_Narayana_row(n, q=None):
return sum([q_Narayana(n, k) for k in (0..n)]).list()
for n in (0..7): print(q_Narayana_row(n))
(Magma)
B:= func< n, k, x | k eq 0 select 1 else (&*[(1-x^(n-j+1))/(1-x^j): j in [1..k]]) >;
QNarayana:= func< n, k, x | B(n, k, x)*B(n-1, k, x)/B(k+1, 1, x) >;
R<x>:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( (&+[QNarayana(n, k, x): k in [0..n]]) )): n in [0..8]]; // G. C. Greubel, May 22 2019
(PARI)
qbin(n, k, x) = prod(j=1, k, (1-x^(n-j+1))/(1-x^j) );
qNarayana(n, k, x) = qbin(n, k, x)*qbin(n-1, k, x)/qbin(k+1, 1, x);
for(n=0, 7, print1(Vecrev(sum(k=0, n, qNarayana(n, n-k, x))), ", "); print(); ) \\ G. C. Greubel, May 23 2019
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Jul 20 2016
STATUS
approved