A274887 Triangle read by rows: coefficients of the q-factorial.

1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1, 1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1

Offset: 0

Author

Peter Luschny, Jul 19 2016

Keywords

• nonn
• tabf

Comments

The main entry for this sequence is A008302 (Mahonian numbers).

q-factorial(n) is a univariate polynomial over the integers with degree n*(n-1)/2.

Evaluated at q=1 the q-factorial(n) gives the factorial A000142(n).

Examples

			The polynomials start:
[0] 1
[1] 1
[2] q + 1
[3] (q + 1) * (q^2 + q + 1)
[4] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1)
[5] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
The triangle starts:
[1]
[1]
[1, 1]
[1, 2, 2, 1]
[1, 3, 5, 6, 5, 3, 1]
[1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1]
[1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1]
		

Links

• G. C. Greubel, Rows n = 0..30 of triangle, flattened
• NIST Digital Library of Mathematical Functions, q-Factorials. (Release 1.0.11 of 2016-06-08)

Crossrefs

Cf. A008302 (the same for all n > 0), A000142 (row sums), A063746 (q-central_binomial), A129175 (q-Catalan), A274886 (q-extended_Catalan), A274888 (q-swing_factorial), A275216 (q-binomial), A275215 (q-Narayana).

Programs

Magma

B:= func< n,x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >;
R:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( B(n,x) )): n in [0..9]]; // G. C. Greubel, May 22 2019

Mathematica

Table[CoefficientList[QFactorial[n,q]//FunctionExpand, q], {n,0,9} ]//Flatten

PARI

for(n=0, 8, print1(Vec(if(n==0, 1, prod(j=1, n, 1-x^j)/(1-x)^n)), ", "); print(); ) \\ G. C. Greubel, May 23 2019

Sage

from sage.combinat.q_analogues import q_factorial
for n in (0..5): print(q_factorial(n).list())

Formula

a(n) = A008302(n) for all n > 0. - M. F. Hasler, Jan 06 2024