A275214 Triangle read by rows, a q-Narayana statistic on Catalan paths.

1, 1, 2, 3, 1, 1, 4, 2, 4, 2, 2, 5, 3, 7, 7, 8, 5, 5, 1, 1, 6, 4, 10, 12, 18, 16, 20, 14, 14, 8, 6, 2, 2, 7, 5, 13, 17, 28, 32, 43, 42, 49, 43, 43, 32, 29, 18, 14, 7, 5, 1, 1, 8, 6, 16, 22, 38, 48, 72, 80, 104, 110, 126, 122, 130, 112, 108, 88, 76, 54, 44, 26, 20, 10, 6, 2, 2

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

Cf. A000108 (row sums), A001263, A275215.

Sequence in context: A023572 A023570 A256989 * A319846 A214690 A238878

Adjacent sequences: A275211 A275212 A275213 * A275215 A275216 A275217

Keyword

nonn,tabf

Author

Peter Luschny, Jul 20 2016

Status

approved