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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:=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.)