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A275214 Triangle read by rows, a q-Narayana statistic on Catalan paths. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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,changed

AUTHOR

Peter Luschny, Jul 20 2016

STATUS

approved

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Last modified May 23 03:04 EDT 2019. Contains 323507 sequences. (Running on oeis4.)