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%I #31 Sep 08 2022 08:46:17
%S 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,
%T 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,
%U 48,72,80,104,110,126,122,130,112,108,88,76,54,44,26,20,10,6,2,2
%N Triangle read by rows, a q-Narayana statistic on Catalan paths.
%C A Catalan path is a Dyck path of length 2n that ends on the horizontal axis.
%H G. C. Greubel, <a href="/A275214/b275214.txt">Rows n = 0..30 of triangle, flattened</a>
%F 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).
%e The polynomials start:
%e 1,
%e 1,
%e 2,
%e 3 + q + q^2,
%e 4 + 2q + 4q^2 + 2q^3 + 2q^4,
%e 5 + 3q + 7q^2 + 7q^3 + 8q^4 + 5q^5 + 5q^6 + q^7 + q^8.
%e The triangle starts:
%e [n] [row] [sum]
%e [0] [1] 1
%e [1] [1] 1
%e [2] [2] 2
%e [3] [3, 1, 1] 5
%e [4] [4, 2, 4, 2, 2] 14
%e [5] [5, 3, 7, 7, 8, 5, 5, 1, 1] 42
%e [6] [6, 4, 10, 12, 18, 16, 20, 14, 14, 8, 6, 2, 2] 132
%t QNarayana[n_,k_]:= QBinomial[n,k,q] QBinomial[n-1,k,q]/QBinomial[k+1,1, q];
%t QNarayanaRow[n_]:= Sum[QNarayana[n,k], {k,0,n}];
%t Table[CoefficientList[QNarayanaRow[n] // FunctionExpand,q], {n,0,8}] // Flatten
%o (Sage)
%o from sage.combinat.q_analogues import q_int, q_binomial
%o def q_Narayana(n, k, q=None):
%o if n == 0: return 1
%o return q_binomial(n,k,q)*q_binomial(n-1,k,q)//q_int(k+1)
%o def q_Narayana_row(n, q=None):
%o return sum([q_Narayana(n, k) for k in (0..n)]).list()
%o for n in (0..7): print(q_Narayana_row(n))
%o (Magma)
%o B:= func< n,k,x | k eq 0 select 1 else (&*[(1-x^(n-j+1))/(1-x^j): j in [1..k]]) >;
%o QNarayana:= func< n,k,x | B(n,k,x)*B(n-1,k,x)/B(k+1,1,x) >;
%o R<x>:=PowerSeriesRing(Integers(), 30);
%o [Coefficients(R!( (&+[QNarayana(n,k,x): k in [0..n]]) )): n in [0..8]]; // _G. C. Greubel_, May 22 2019
%o (PARI)
%o qbin(n, k, x) = prod(j=1, k, (1-x^(n-j+1))/(1-x^j) );
%o qNarayana(n, k, x) = qbin(n,k,x)*qbin(n-1,k,x)/qbin(k+1,1,x);
%o for(n=0, 7, print1(Vecrev(sum(k=0,n, qNarayana(n,n-k,x))), ", "); print(); ) \\ _G. C. Greubel_, May 23 2019
%Y Cf. A000108 (row sums), A001263, A275215.
%K nonn,tabf
%O 0,3
%A _Peter Luschny_, Jul 20 2016
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