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A008967
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Triangle of coefficients of Gaussian polynomials [ n,2 ]; also triangle of distribution of rank sums: Wilcoxon's statistic.
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6
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1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 4, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 4,7
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COMMENTS
| Rows are numbers of dominoes with k spots where each half-domino has zero to n spots (in standard domino set: n=6, there are 28 dominoes and row is 1,1,2,2,3,3,4,3,3,2,2,1,1) - Henry Bottomley (se16(AT)btinternet.com), Aug 23 2000
The Gaussian polynomial (or Gaussian binomial) [n,2]_q is an example of a q-binomial coefficient (see the link) and may be defined for n >= 2 by [n,2]_q = ([n]_q * [n-1]_q)/([1]_q * [2]_q), where [n]_q := q^n - 1. The first few values are: [2,2]_q = 1; [3,2]_q = 1 + q + q^2; [4,2]_q = 1 + q + 2q^2 + q^3 + q^4. - 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.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 236.
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LINKS
| Eric Weisstein's World of Mathematics, q-Binomial Coefficient.
Index entries for sequences related to dominoes
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FORMULA
| Let f(r) = Product( (x^i-x^(r+1))/(1-x^i), i = 1..r-2) / x^((r-1)*(r-2)/2); then expanding f(r) in powers of x and taking coefficients gives the successive rows of this triangle (with a different offset).
Expanding (q^n - 1)(q^(n+1) - 1)/((q - 1)(q^2 - 1)) in powers of q and taking coefficients gives the n th row of the triangle. Ordinary generating function: 1/((1-x)(1-qx)(1-q^2x)) = 1 + x(1 + q + q^2) + x^2(1 + q + 2q^2 + q^3 + q^4) + .... - Peter Bala (pbala(AT)toucansurf.com), Sep 23 2007
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EXAMPLE
| 1;
1,1,1;
1,1,2,1,1;
1,1,2,2,2,1,1;
1,1,2,2,3,2,2,1,1;
1,1,2,2,3,3,3,2,2,1,1;
...
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MAPLE
| qBinom := proc(n, m, q)
mul((1-q^(n-i))/(1-q^(i+1)), i=0..m-1) ;
factor(%) ;
expand(%) ;
end proc:
A008967 := proc(n, k)
coeftayl( qBinom(n, 2, q), q=0, k ) ;
end proc:
seq(seq( A008967(n, k), k=0..2*n-4), n=2..10) ; # assumes offset 2. R. J. Mathar, Oct 13 2011
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MATHEMATICA
| rmax = 11; f[r_] := Product[(x^i - x^(r+1))/(1-x^i), {i, 1, r-2}]/ x^((r-1)*(r-2)/2); row[r_] := CoefficientList[ Series[ f[r], {x, 0, 2rmax}], x]; Flatten[ Table[ row[r], {r, 2, rmax}]] (* From Jean-François Alcover, Oct 13 2011, after given formula *)
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CROSSREFS
| Cf. A047971, A008968, A106822.
Sequence in context: A002635 A108244 A124961 * A094189 A122771 A112190
Adjacent sequences: A008964 A008965 A008966 * A008968 A008969 A008970
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KEYWORD
| tabf,nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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