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A008967 Triangle of coefficients of Gaussian polynomials [ n+2,2 ]; also triangle of distribution of rank sums: Wilcoxon's statistic. 10
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; text; internal format)
OFFSET

4,7

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, 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, Sep 23 2007

These numbers appear in the solution of Cayley's counting problem on covariants as N(p,2,w) = [x^p,q^w] Phi(q,x) with the o.g.f. Phi(q,x) = 1/((1-x)(1-qx)(1-q^2x)) given by Peter Bala in the formula section. See the Hawkins reference, p. 264, were also references are given. - Wolfdieter Lang, Nov 30 2012

The entry a(p,w), p >= 0, w = 0,1,...,2*p, of this irregular triangle is the number of nonnegative solutions of m_0 + m_1 + m_2 = p and 1*m_1 + 2*m_2 = w. See the Hawkins reference p. 264, (4.8). N(p,2,w) there is a(p,w). See also the Cayley reference p. 110, 35. with m = 2, Theta = p and q = w. - Wolfdieter Lang, Dec 01 2012

REFERENCES

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 242.

A. Cayley, A Second Memoir Upon Quantics, Phil. Trans. R. Soc. London, 146 (1856) 101 - 126.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 236.

T. Hawkins, Emergence of the Theory of Lie Groups, Springer 2000, ch. 7.4, p. 260-5.

LINKS

Table of n, a(n) for n=4..97.

Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Index entries for sequences related to dominoes

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, Sep 23 2007

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;

...

Partitions: row p=2 and column w=2 has entry 2 because the 2 solutions of the two equations mentioned in a comment above are: m_0 = 0, m_1 = 2, m_2 = 0 and m_0 = 1, m_1 = 0, m_2 = 1. - Wolfdieter Lang, Dec 01 2012

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

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}]] (* Jean-Fran├žois Alcover, Oct 13 2011, after given formula *)

CROSSREFS

Cf. A008968, A047971, A106822.

Sequence in context: A277824 A265120 A124961 * A211355 A211353 A094189

Adjacent sequences:  A008964 A008965 A008966 * A008968 A008969 A008970

KEYWORD

tabf,nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)