%I #130 Sep 09 2024 02:50:44
%S 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,
%T 63,1,1,127,2667,11811,11811,2667,127,1,1,255,10795,97155,200787,
%U 97155,10795,255,1,1,511,43435,788035,3309747,3309747,788035,43435,511,1
%N Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2.
%C Also number of distinct binary linear [n,k] codes.
%C Row sums give A006116.
%C Central terms are A006098.
%C T(n,k) is the number of subgroups of the Abelian group (C_2)^n that have order 2^k. - _Geoffrey Critzer_, Mar 28 2016
%C T(n,k) is the number of k-subspaces of the finite vector space GF(2)^n. - _Jianing Song_, Jan 31 2020
%D J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
%D F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%D M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H T. D. Noe, <a href="/A022166/b022166.txt">Rows n=0..50 of triangle, flattened</a>
%H Octavio A. Agustín-Aquino, <a href="http://arxiv.org/abs/1602.05279">Archimedes' quadrature of the parabola and minimal covers</a>, arXiv:1602.05279 [math.CO], 2016.
%H J. A. de Azcarraga and J. A. Macfarlane, <a href="http://arxiv.org/abs/hep-th/9506177">Group Theoretical Foundations of Fractional Supersymmetry</a> arXiv:hep-th/9506177, 1995.
%H Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.
%H Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, <a href="https://arxiv.org/abs/2012.13570">Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow</a>, arXiv:2012.13570 [math.CO], 2020.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv:1409.3820 [math.NT], 2014.
%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H D. Slepian, <a href="https://archive.org/details/bstj35-1-203">A class of binary signaling alphabets</a>, Bell System Tech. J. 35 (1956), 203-234.
%H D. Slepian, <a href="https://archive.org/details/bstj39-5-1219">Some further theory of group codes</a>, Bell System Tech. J. 39 1960 1219-1252.
%H M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Q-binomial">q-binomial</a>
%H <a href="/index/Coa#codes_binary_linear">Index entries for sequences related to binary linear codes</a>
%H <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries for sequences related to Gaussian binomial coefficients</a>
%F G.f.: A(x,y) = Sum_{k>=0} y^k/Product_{j=0..k} (1 - 2^j*x). - _Paul D. Hanna_, Oct 28 2006
%F For k = 1,2,3,... the expansion of exp( Sum_{n >= 1} (2^(k*n) - 1)/(2^n - 1)*x^n/n ) gives the o.g.f. for the k-th diagonal of the triangle (k = 1 corresponds to the main diagonal). - _Peter Bala_, Apr 07 2015
%F T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - _Peter A. Lawrence_, Jul 13 2017
%F T(m+n,k) = Sum_{i=0..k} q^((k-i)*(m-i)) * T(m,i) * T(n,k-i), q=2 (see the Sved link, page 337). - _Werner Schulte_, Apr 09 2019
%F T(n,k) = Sum_{j=0..k} qStirling2(n-j,n-k)*C(n,j) where qStirling2(n,k) is A139382. - _Vladimir Kruchinin_, Mar 04 2020
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 7, 7, 1;
%e 1, 15, 35, 15, 1;
%e 1, 31, 155, 155, 31, 1;
%e 1, 63, 651, 1395, 651, 63, 1;
%e 1, 127, 2667, 11811, 11811, 2667, 127, 1;
%p A005329 := proc(n)
%p mul( 2^i-1,i=1..n) ;
%p end proc:
%p A022166 := proc(n,m)
%p A005329(n)/A005329(n-m)/A005329(m) ;
%p end proc: # _R. J. Mathar_, Nov 14 2011
%t Table[QBinomial[n, k, 2], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 08 2016 *)
%t (* S stands for qStirling2 *) S[n_, k_, q_] /; 1 <= k <= n := S[n - 1, k - 1, q] + Sum[q^j, {j, 0, k - 1}]*S[n - 1, k, q]; S[n_, 0, _] := KroneckerDelta[n, 0]; S[0, k_, _] := KroneckerDelta[0, k]; S[_, _, _] = 0;
%t T[n_, k_] /; n >= k := Sum[Binomial[n, j]*S[n - j, n - k, q]*(q - 1)^(k - j) /. q -> 2, {j, 0, k}];
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 08 2020, after _Vladimir Kruchinin_ *)
%o (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
%o (PARI) qp = matpascal(9,2);
%o for(n=1,#qp,for(k=1,n,print1(qp[n,k],", "))) \\ _Gerald McGarvey_, Dec 05 2009
%o (PARI) {q=2; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || n<k, 0, T(n-1,k-1) + q^k*T(n-1,k))))};
%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018
%o (Sage) def T(n,k): return gaussian_binomial(n,k).subs(q=2) # _Ralf Stephan_, Mar 02 2014
%o (Magma) q:=2; [[k le 0 select 1 else (&*[(1-q^(n-j))/(1-q^(j+1)): j in [0..(k-1)]]): k in [0..n]]: n in [0..20]]; // _G. C. Greubel_, Nov 17 2018
%Y Cf. A006516, A218449, A135950 (matrix inverse), A000225 (k=1), A006095 (k=2), A006096 (k=3), A139382.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_