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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<x>:=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 A171414 A270921

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 June 26 00:10 EDT 2019. Contains 324367 sequences. (Running on oeis4.)