Search: id:A008967
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%I A008967
%S A008967 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,
%T A008967 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,
%U A008967 4,4,5,4,4,3,3,2,2,1,1,1,1,2,2,3,3,4,4,5,5,5,4,4
%N A008967 Triangle of coefficients of Gaussian polynomials [ n,2 ]; also triangle 
               of distribution of rank sums: Wilcoxon's statistic.
%C A008967 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
%C A008967 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
%D A008967 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 242.
%D A008967 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 236.
%H A008967 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               q-BinomialCoefficient.html">q-Binomial Coefficient</a>.
%H A008967 <a href="Sindx_Do.html#domino">Index entries for sequences related to 
               dominoes</a>
%F A008967 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).
%F A008967 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
%e A008967 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; ...
%Y A008967 Cf. A047971, A008968, A106822.
%Y A008967 Sequence in context: A002635 A108244 A124961 this_sequence A094189 A122771 
               A112188
%Y A008967 Adjacent sequences: A008964 A008965 A008966 this_sequence A008968 A008969 
               A008970
%K A008967 tabf,nonn,nice,easy
%O A008967 4,7
%A A008967 N. J. A. Sloane (njas(AT)research.att.com).

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