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 A022166 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2. 67
 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, 3309747, 3309747, 788035, 43435, 511, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also number of distinct binary linear [n,k] codes. Row sums give A006116. Central terms are A006098. 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 T(n,k) is the number of k-subspaces of the finite vector space GF(2)^n. - Jianing Song, Jan 31 2020 REFERENCES 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. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698. M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. LINKS T. D. Noe, Rows n=0..50 of triangle, flattened Octavio Alberto Agustín Aquino, Archimedes' quadrature of the parabola and minimal covers, arXiv:1602.05279 [math.CO], 2016. J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations of Fractional Supersymmetry arXiv:hep-th/9506177, 1995. Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018. R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv:1409.3820 [math.NT], 2014. Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1. 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. M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy) Eric W. Weisstein, q-Binomial Coefficient. Wikipedia, q-binomial 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 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 T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017 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 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 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; MAPLE A005329 := proc(n)    mul( 2^i-1, i=1..n) ; end proc: A022166 := proc(n, m)    A005329(n)/A005329(n-m)/A005329(m) ; end proc: # R. J. Mathar, Nov 14 2011 MATHEMATICA Table[QBinomial[n, k, 2], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 08 2016 *) (* 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[n_, k_] /; n >= k := Sum[Binomial[n, j]*S[n - j, n - k, q]*(q - 1)^(k - j) /. q -> 2, {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 08 2020, after Vladimir Kruchinin *) 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) qp = matpascal(9, 2); for(n=1, #qp, for(k=1, n, print1(qp[n, k], ", "))) \\ Gerald McGarvey, Dec 05 2009 (PARI) {q=2; T(n, k) = if(k==0, 1, if (k==n, 1, if (k<0 || n

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Last modified September 24 20:13 EDT 2020. Contains 337321 sequences. (Running on oeis4.)