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A008967 Coefficients of Gaussian polynomials q_binomial(n-2, 2). Also triangle of distribution of rank sums: Wilcoxon's statistic. Irregular triangle read by rows. 11
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, 3, 3, 2, 2, 1, 1 (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.

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..103.

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

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

For n >= 2, let a(n,i) denote the i-th entry of the (n-1)-st row of this triangle; for every 0 <= i <= n-2, a(n,i) = a(n,2(n-2)-i) = ceiling((i+1)/2). - Christian Barrientos, Aug 08 2019

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-François Alcover, Oct 13 2011, after given formula *)

T[n_, k_] := SeriesCoefficient[QBinomial[n - 2, 2, q], {q, 0, k}];

Table[T[n, k], {n, 4, 13}, {k, 0, 2 n - 8}] // Flatten (* Jean-François Alcover, Aug 20 2019 *)

PROG

(SageMath)

print(flatten([q_binomial(n-2, 2).list() for n in (4..13)])) # Peter Luschny, Oct 23 2019

CROSSREFS

Cf. A008968, A047971, A106822.

Sequence in context: A265120 A329621 A124961 * A211355 A211353 A094189

Adjacent sequences:  A008964 A008965 A008966 * A008968 A008969 A008970

KEYWORD

tabf,nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Christian Barrientos, Aug 08 2019

STATUS

approved

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Last modified May 26 09:37 EDT 2020. Contains 334620 sequences. (Running on oeis4.)