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.
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, 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, 2, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1
Offset: 0
Examples
The polynomials start: [0] 1 [1] q + 1 [2] q^2 + q + 1 [3] (q + 1) * (q^2 + 1) * (q^2 + q + 1) [4] (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1) [5] (q + 1)*(q^2 - q + 1)*(q^2 + 1)*(q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1) Triangle starts: [0] [1] [1] [1, 1] [2] [1, 1, 1] [3] [1, 2, 3, 3, 2, 1] [4] [1, 1, 2, 2, 2, 1, 1] [5] [1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1] [6] [1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1] [7] [1, 2, 4, 7, 11, 15, 20, 24, 27, 29, 29, 27, 24, 20, 15, 11, 7, 4, 2, 1]
Links
- G. C. Greubel, Rows n = 0..35 of triangle, flattened
- Peter Luschny, Orbitals
Programs
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Magma
QFac:= func< n, x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >; P:= func< n,x | QFac(n+1,x)/( QFac(Floor(n/2),x)*QFac(Floor((n+2)/2),x) ) >; R
:=PowerSeriesRing(Integers(), 30); [Coefficients(R!( P(n,x) )): n in [0..8]]; // G. C. Greubel, May 22 2019 -
Maple
Qbinom1 := proc(n) local F, h; h := iquo(n,2); F := x -> QDifferenceEquations:-QFactorial(x,q); F(n+1)/(F(h)*F(h+1)); expand(simplify(expand(%))); seq(coeff(%,q,j), j=0..degree(%)) end: seq(Qbinom1(n), n=0..8);
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Mathematica
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 -
Sage
from sage.combinat.q_analogues import q_factorial def q_binom1(n): return (q_factorial(n+1)//(q_factorial(n//2)* q_factorial((n+2)//2))) for n in (0..10): print(q_binom1(n).list())