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A274887 Triangle read by rows: coefficients of the q-factorial.

%I A274887 #33 Jan 06 2024 11:28:42
%S A274887 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,
%T A274887 49,71,90,101,101,90,71,49,29,14,5,1,1,6,20,49,98,169,259,359,455,531,
%U A274887 573,573,531,455,359,259,169,98,49,20,6,1
%N A274887 Triangle read by rows: coefficients of the q-factorial.
%C A274887 The main entry for this sequence is A008302 (Mahonian numbers).
%C A274887 q-factorial(n) is a univariate polynomial over the integers with degree n*(n-1)/2.
%C A274887 Evaluated at q=1 the q-factorial(n) gives the factorial A000142(n).
%H A274887 G. C. Greubel, <a href="/A274887/b274887.txt">Rows n = 0..30 of triangle, flattened</a>
%H A274887 NIST Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/5.18#i">q-Factorials</a>. (Release 1.0.11 of 2016-06-08)
%F A274887 a(n) = A008302(n) for all n > 0. - _M. F. Hasler_, Jan 06 2024
%e A274887 The polynomials start:
%e A274887 [0] 1
%e A274887 [1] 1
%e A274887 [2] q + 1
%e A274887 [3] (q + 1) * (q^2 + q + 1)
%e A274887 [4] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1)
%e A274887 [5] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
%e A274887 The triangle starts:
%e A274887 [1]
%e A274887 [1]
%e A274887 [1, 1]
%e A274887 [1, 2, 2, 1]
%e A274887 [1, 3, 5, 6, 5, 3, 1]
%e A274887 [1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1]
%e A274887 [1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1]
%t A274887 Table[CoefficientList[QFactorial[n,q]//FunctionExpand, q], {n,0,9} ]//Flatten
%o A274887 (Sage)
%o A274887 from sage.combinat.q_analogues import q_factorial
%o A274887 for n in (0..5): print(q_factorial(n).list())
%o A274887 (Magma)
%o A274887 B:= func< n,x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >;
%o A274887 R<x>:=PowerSeriesRing(Integers(), 30);
%o A274887 [Coefficients(R!( B(n,x) )): n in [0..9]]; // _G. C. Greubel_, May 22 2019
%o A274887 (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
%Y A274887 Cf. A008302 (the same for all n > 0), A000142 (row sums), A063746 (q-central_binomial), A129175 (q-Catalan), A274886 (q-extended_Catalan), A274888 (q-swing_factorial), A275216 (q-binomial), A275215 (q-Narayana).
%K A274887 nonn,tabf
%O A274887 0,6
%A A274887 _Peter Luschny_, Jul 19 2016