Search: id:A022166
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%I A022166
%S A022166 1,1,1,1,3,1,1,7,7,1,1,15,35,15,1,1,31,155,155,31,1,1,63,
%T A022166 651,1395,651,63,1,1,127,2667,11811,11811,2667,127,1,1,
%U A022166 255,10795,97155,200787,97155,10795,255,1,1,511,43435,788035
%N A022166 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) 
               [n,k] for q = 2.
%C A022166 Also number of distinct binary linear [n,k] codes.
%D A022166 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting 
               Codes, Elsevier-North Holland, 1978, p. 698.
%D A022166 Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, 
               Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%D A022166 D. Slepian, A class of binary signaling alphabets. Bell System Tech. 
               J. 35 (1956), 203-234.
%D A022166 D. Slepian, Some further theory of group codes. Bell System Tech. J. 
               39 1960 1219-1252.
%D A022166 J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations 
               of Fractional Supersymmetry ,- Arxiv preprint hep-th/9506177, 1995 
               - arxiv.org: http://arxiv.org/abs/hep-th/9506177 [From Roger L. Bagula 
               (rlbagulatftn(AT)yahoo.com), Feb 09 2009]
%H A022166 T. D. Noe, <a href="b022166.txt">Rows n=0..50 of triangle, flattened</
               a>
%H A022166 <a href="Sindx_Coa.html#codes_binary_linear">Index entries for sequences 
               related to binary linear codes</a>
%F A022166 G.f.: A(x,y) = Sum_{k>=0} y^k/Product_{j=0..k} (1 - 2^j*x). - Paul D. 
               Hanna (pauldhanna(AT)juno.com), Oct 28 2006
%F A022166 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 09 
               2009: (Start)
%F A022166 Definition of m here is q-1:
%F A022166 t(n,m)=If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
%F A022166 C(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])] (End)
%e A022166 Triangle begins:
%e A022166 1;
%e A022166 1, 1;
%e A022166 1, 3, 1;
%e A022166 1, 7, 7, 1;
%e A022166 1, 15, 35, 15, 1;
%e A022166 1, 31, 155, 155, 31, 1;
%e A022166 1, 63, 651, 1395, 651, 63, 1;
%e A022166 1, 127, 2667, 11811, 11811, 2667, 127, 1;
%t A022166 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 09 
               2009: (Start)
%t A022166 Clear[t, n, m, i, k, a, b];
%t A022166 t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 
               1, n}]];
%t A022166 b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
%t A022166 c = Table[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];
%t A022166 TableForm[c];
%t A022166 (*A022166, A022167, A022168, A022169, A022170, A022171, A022175*)
%t A022166 Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 
               15}] (End)
%o A022166 (PARI) T(n,k)=polcoeff(x^k/prod(j=0,k,1-2^j*x+x*O(x^n)),n) - Paul D. 
               Hanna (pauldhanna(AT)juno.com), Oct 28 2006
%Y A022166 Row sums give A006116. Cf. A006516, A022167.
%Y A022166 Central terms are A006098.
%Y A022166 A022166, A022167, A022168, A022169, A022170, A022171, A022175 [From Roger 
               L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 09 2009]
%Y A022166 Sequence in context: A157152 A136126 A046802 this_sequence A141689 A058669 
               A057004
%Y A022166 Adjacent sequences: A022163 A022164 A022165 this_sequence A022167 A022168 
               A022169
%K A022166 nonn,tabl
%O A022166 0,5
%A A022166 N. J. A. Sloane (njas(AT)research.att.com).

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