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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. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS G. C. Greubel, Rows n = 0..35 of triangle, flattened Peter Luschny, Orbitals EXAMPLE 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] 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); 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 PROG (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()) (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 CROSSREFS Cf. Row sums are A212303(n+1) and A275212(n,0), A274886. Sequence in context: A261283 A123548 A131838 * A287732 A334223 A171414 Adjacent sequences: A274882 A274883 A274884 * A274886 A274887 A274888 KEYWORD nonn,tabf AUTHOR Peter Luschny, Jul 20 2016 STATUS approved

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Last modified February 7 07:00 EST 2023. Contains 360112 sequences. (Running on oeis4.)