

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
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OFFSET

4,7


COMMENTS

Rows are numbers of dominoes with k spots where each halfdomino 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 qbinomial coefficient (see the link) and may be defined for n >= 2 by [n,2]_q = ([n]_q * [n1]_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/((1x)(1qx)(1q^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, AddisonWesley, 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. 2605.


LINKS

Table of n, a(n) for n=4..97.
Eric Weisstein's World of Mathematics, qBinomial Coefficient.
Index entries for sequences related to dominoes


FORMULA

Let f(r) = Product( (x^ix^(r+1))/(1x^i), i = 1..r2) / x^((r1)*(r2)/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/((1x)(1qx)(1q^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((1q^(ni))/(1q^(i+1)), i=0..m1) ;
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*n4), n=2..10) ; # assumes offset 2. R. J. Mathar, Oct 13 2011


MATHEMATICA

rmax = 11; f[r_] := Product[(x^i  x^(r+1))/(1x^i), {i, 1, r2}]/ x^((r1)*(r2)/2); row[r_] := CoefficientList[ Series[ f[r], {x, 0, 2rmax}], x]; Flatten[ Table[ row[r], {r, 2, rmax}]] (* JeanFrançois 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



