A274885 - OEIS

%I #24 Sep 08 2022 08:46:17

%S 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 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 2,1,1,1,2,3,5,6,8,9,11,11,12,11,11,9,8,6,5,3,2,1,1

%N 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 G. C. Greubel, <a href="/A274885/b274885.txt">Rows n = 0..35 of triangle, flattened</a>

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>

%e The polynomials start:

%e [0] 1

%e [1] q + 1

%e [2] q^2 + q + 1

%e [3] (q + 1) * (q^2 + 1) * (q^2 + q + 1)

%e [4] (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1)

%e [5] (q + 1)*(q^2 - q + 1)*(q^2 + 1)*(q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)

%e Triangle starts:

%e [0] [1]

%e [1] [1, 1]

%e [2] [1, 1, 1]

%e [3] [1, 2, 3, 3, 2, 1]

%e [4] [1, 1, 2, 2, 2, 1, 1]

%e [5] [1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1]

%e [6] [1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1]

%e [7] [1, 2, 4, 7, 11, 15, 20, 24, 27, 29, 29, 27, 24, 20, 15, 11, 7, 4, 2, 1]

%p Qbinom1 := proc(n) local F, h; h := iquo(n,2);

%p F := x -> QDifferenceEquations:-QFactorial(x,q);

%p F(n+1)/(F(h)*F(h+1)); expand(simplify(expand(%)));

%p seq(coeff(%,q,j), j=0..degree(%)) end: seq(Qbinom1(n), n=0..8);

%t 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 (Sage)

%o from sage.combinat.q_analogues import q_factorial

%o def q_binom1(n): return (q_factorial(n+1)//(q_factorial(n//2)* q_factorial((n+2)//2)))

%o for n in (0..10): print(q_binom1(n).list())

%o (Magma)

%o QFac:= func< n, x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >;

%o P:= func< n,x | QFac(n+1,x)/( QFac(Floor(n/2),x)*QFac(Floor((n+2)/2),x) ) >;

%o R<x>:=PowerSeriesRing(Integers(), 30);

%o [Coefficients(R!( P(n,x) )): n in [0..8]]; // _G. C. Greubel_, May 22 2019

%Y Cf. Row sums are A212303(n+1) and A275212(n,0), A274886.

%K nonn,tabf

%O 0,8

%A _Peter Luschny_, Jul 20 2016