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A274885 Coefficients of some q-polynomials, P_n(q) = q_factorial(n+1) / (q_factorial([n/2]) * q_factorial([(n+2)/2])) with [.] the floor function.

%I A274885 #24 Sep 08 2022 08:46:17
%S A274885 1,1,1,1,1,1,1,2,3,3,2,1,1,1,2,2,2,1,1,1,2,4,6,8,9,9,8,6,4,2,1,1,1,2,
%T A274885 3,4,4,5,4,4,3,2,1,1,1,2,4,7,11,15,20,24,27,29,29,27,24,20,15,11,7,4,
%U A274885 2,1,1,1,2,3,5,6,8,9,11,11,12,11,11,9,8,6,5,3,2,1,1
%N A274885 Coefficients of some q-polynomials, P_n(q) = q_factorial(n+1) / (q_factorial([n/2]) * q_factorial([(n+2)/2])) with [.] the floor function.
%H A274885 G. C. Greubel, <a href="/A274885/b274885.txt">Rows n = 0..35 of triangle, flattened</a>
%H A274885 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e A274885 The polynomials start:
%e A274885 [0] 1
%e A274885 [1] q + 1
%e A274885 [2] q^2 + q + 1
%e A274885 [3] (q + 1) * (q^2 + 1) * (q^2 + q + 1)
%e A274885 [4] (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1)
%e A274885 [5] (q + 1)*(q^2 - q + 1)*(q^2 + 1)*(q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
%e A274885 Triangle starts:
%e A274885 [0] [1]
%e A274885 [1] [1, 1]
%e A274885 [2] [1, 1, 1]
%e A274885 [3] [1, 2, 3, 3, 2, 1]
%e A274885 [4] [1, 1, 2, 2, 2, 1, 1]
%e A274885 [5] [1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1]
%e A274885 [6] [1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1]
%e A274885 [7] [1, 2, 4, 7, 11, 15, 20, 24, 27, 29, 29, 27, 24, 20, 15, 11, 7, 4, 2, 1]
%p A274885 Qbinom1 := proc(n) local F, h; h := iquo(n,2);
%p A274885 F := x -> QDifferenceEquations:-QFactorial(x,q);
%p A274885 F(n+1)/(F(h)*F(h+1)); expand(simplify(expand(%)));
%p A274885 seq(coeff(%,q,j), j=0..degree(%)) end: seq(Qbinom1(n), n=0..8);
%t A274885 QBinom1[n_] := QFactorial[n+1,q] / (QFactorial[Quotient[n,2],q] QFactorial[Quotient[n+2,2],q]); Table[CoefficientList[QBinom1[n] // FunctionExpand,q], {n,0,8}] // Flatten
%o A274885 (Sage)
%o A274885 from sage.combinat.q_analogues import q_factorial
%o A274885 def q_binom1(n): return (q_factorial(n+1)//(q_factorial(n//2)* q_factorial((n+2)//2)))
%o A274885 for n in (0..10): print(q_binom1(n).list())
%o A274885 (Magma)
%o A274885 QFac:= func< n, x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >;
%o A274885 P:= func< n,x | QFac(n+1,x)/( QFac(Floor(n/2),x)*QFac(Floor((n+2)/2),x) ) >;
%o A274885 R<x>:=PowerSeriesRing(Integers(), 30);
%o A274885 [Coefficients(R!( P(n,x) )): n in [0..8]]; // _G. C. Greubel_, May 22 2019
%Y A274885 Cf. Row sums are A212303(n+1) and A275212(n,0), A274886.
%K A274885 nonn,tabf
%O A274885 0,8
%A A274885 _Peter Luschny_, Jul 20 2016