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 A274887 Triangle read by rows: coefficients of the q-factorial. 3
 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1, 1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS The main entry for this sequence is A008302 (Mahonian numbers). q-factorial(n) is a univariate polynomial over the integers with degree n*(n-1)/2. Evaluated at q=1 the q-factorial(n) gives the factorial A000142(n). LINKS G. C. Greubel, Rows n = 0..30 of triangle, flattened NIST Digital Library of Mathematical Functions, q-Factorials. (Release 1.0.11 of 2016-06-08) EXAMPLE The polynomials start: [0] 1 [1] 1 [2] q + 1 [3] (q + 1) * (q^2 + q + 1) [4] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1) [5] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1) The triangle starts: [1] [1] [1, 1] [1, 2, 2, 1] [1, 3, 5, 6, 5, 3, 1] [1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1] [1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1] MATHEMATICA Table[CoefficientList[QFactorial[n, q]//FunctionExpand, q], {n, 0, 9} ]//Flatten PROG (Sage) from sage.combinat.q_analogues import q_factorial for n in (0..5): print q_factorial(n).list() (MAGMA) B:= func< n, x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >; R:=PowerSeriesRing(Integers(), 30); [Coefficients(R!( B(n, x) )): n in [0..9]]; // G. C. Greubel, May 22 2019 (PARI) for(n=0, 8, print1(Vec(if(n==0, 1, prod(j=1, n, 1-x^j)/(1-x)^n)), ", "); print(); ) \\ G. C. Greubel, May 23 2019 CROSSREFS Cf. A008302, A000142 (row sums), A063746 (q-central_binomial), A129175 (q-Catalan), A274886 (q-extended_Catalan), A274888 (q-swing_factorial), A275216 (q-binomial), A275215 (q-Narayana). Sequence in context: A215563 A076263 A272689 * A008302 A131791 A308497 Adjacent sequences:  A274884 A274885 A274886 * A274888 A274889 A274890 KEYWORD nonn,tabf AUTHOR Peter Luschny, Jul 19 2016 STATUS approved

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Last modified June 16 12:38 EDT 2019. Contains 324152 sequences. (Running on oeis4.)