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A047971
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Triangle of coefficients of Gaussian polynomials [ n,3 ].
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2
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1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 5, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3, 2, 1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| a(n) as illustrated is related to the following sequences: The row sum values are A001400. The column sums are A000292. The row lengths are the stuttering sequence A037915 (stutter values in A016777). The column lengths are the sequence A016777. The max values in each column are A001971. - Alford Arnold (Alford1940(AT)aol.com), Aug 16 2004
The Gaussian polynomial (or Gaussian binomial) [n,3]_q is an example of a q-binomial coefficient (see the link) and may be defined for n >= 3 by [n,3]_q = ([n]_q * [n-1]_q * [n-2]_q)/([1]_q * [2]_q * [3]_q), where [n]_q := q^n - 1. The first few values are: [3,3]_q = 1; [4,3]_q = 1 + q + q^2 + q^3; [5,3]_q = 1 + q + 2q^2 + 2q^3 + 2q^4 + q^5 + q^6. - Peter Bala (pbala(AT)toucansurf.com), Sep 23 2007
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REFERENCES
| G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 242.
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LINKS
| Eric Weisstein's World of Mathematics, q-Binomial Coefficient.
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FORMULA
| O.g.f.: 1/((1-x)(1-qx)(1-q^2x)(1-q^3x)) = 1 + x(1 + q + q^2 + q^3) + x^2(1 + q + 2q^2 + 2q^3 + 2q^4 + q^5 + q^6) + .... - Peter Bala (pbala(AT)toucansurf.com), Sep 23 2007
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EXAMPLE
| 1; 1 1 1 1; 1 1 2 2 2 1 1; 1 1 2 3 3 3 3 2 1 1; ...
The table may also be arranged as follows:
1
..1
..1..1
..1..1..1
..1..2..1..1
.....2..2..1..1
.....2..3..2..1..1
.....1..3..3..2..1..1
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MATHEMATICA
| nmax = 6; se = Series[ 1/Product[1 - q^k*x, {k, 0, 3}], {x, 0, nmax}]; row[n_] := CoefficientList[ SeriesCoefficient[se, n], q]; Flatten[ Table[ row[n], {n, 0, nmax}]] (* From Jean-François Alcover, Dec 19 2011 *)
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CROSSREFS
| Cf. A008967.
Cf. A005400.
Sequence in context: A037804 A081503 A176508 * A029432 A073426 A199840
Adjacent sequences: A047968 A047969 A047970 * A047972 A047973 A047974
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KEYWORD
| nonn,easy,nice,tabf
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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