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A022166
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Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2.
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17
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1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 35, 15, 1, 1, 31, 155, 155, 31, 1, 1, 63, 651, 1395, 651, 63, 1, 1, 127, 2667, 11811, 11811, 2667, 127, 1, 1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1, 1, 511, 43435, 788035
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Also number of distinct binary linear [n,k] codes.
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REFERENCES
| F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
D. Slepian, A class of binary signaling alphabets. Bell System Tech. J. 35 (1956), 203-234.
D. Slepian, Some further theory of group codes. Bell System Tech. J. 39 1960 1219-1252.
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LINKS
| T. D. Noe, Rows n=0..50 of triangle, flattened
J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations of Fractional Supersymmetry arXiv:hep-th/9506177
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Weisstein, Eric W. q-Binomial Coefficient from Mathworld.
Wikipedia, q-binomial
Index entries for sequences related to binary linear codes
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FORMULA
| G.f.: A(x,y) = Sum_{k>=0} y^k/Product_{j=0..k} (1 - 2^j*x). - Paul D. Hanna, Oct 28 2006
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EXAMPLE
| Triangle begins:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 35, 15, 1;
1, 31, 155, 155, 31, 1;
1, 63, 651, 1395, 651, 63, 1;
1, 127, 2667, 11811, 11811, 2667, 127, 1;
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MAPLE
| A005329 := proc(n)
mul( 2^i-1, i=1..n) ;
end proc:
A022166 := proc(n, k)
A005329(n)/A005329(n-m)/A005329(m) ;
end proc: # R. J. Mathar, Nov 14 2011
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MATHEMATICA
| Contribution from Roger L. Bagula, Feb 09 2009: (Start)
t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
c = Table[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];
TableForm[c];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}] (End)
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PROG
| (PARI) T(n, k)=polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n) /* Paul D. Hanna, Oct 28 2006 */
(PARI) s = 9; q = 2; qp = matpascal(s, q);
for(n=1, s+1, for(k=1, n, print1(qp[n, k], ", ")) /* Gerald McGarvey, Dec 05 2009 */
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CROSSREFS
| Row sums give A006116. Cf. A006516, A022167 - A022171, A022175.
Central terms are A006098.
Sequence in context: A136126 A046802 A184173 * A141689 A058669 A057004
Adjacent sequences: A022163 A022164 A022165 * A022167 A022168 A022169
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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