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A274887
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Triangle read by rows: coefficients of the q-factorial.
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3
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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
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OFFSET
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0,6
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COMMENTS
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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).
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LINKS
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NIST Digital Library of Mathematical Functions, q-Factorials. (Release 1.0.11 of 2016-06-08)
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FORMULA
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EXAMPLE
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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]
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MATHEMATICA
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Table[CoefficientList[QFactorial[n, q]//FunctionExpand, q], {n, 0, 9} ]//Flatten
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PROG
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(Sage)
from sage.combinat.q_analogues import q_factorial
for n in (0..5): print(q_factorial(n).list())
(Magma)
B:= func< n, x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >;
R<x>:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( B(n, x) )): n in [0..9]]; // G. C. Greubel, May 22 2019
(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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