A022166 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2.

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

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 A. 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.

Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.

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

Index entries for sequences related to binary linear codes

Index entries for sequences related to Gaussian binomial coefficients

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<k, 0, T(n-1, k-1) + q^k*T(n-1, k))))};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 27 2018
(Sage) def T(n, k): return gaussian_binomial(n, k).subs(q=2) # Ralf Stephan, Mar 02 2014
(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

Crossrefs

Cf. A006516, A218449, A135950 (matrix inverse), A000225 (k=1), A006095 (k=2), A006096 (k=3), A139382.

Sequence in context: A046802 A184173 A359985 * A141689 A058669 A057004

Adjacent sequences: A022163 A022164 A022165 * A022167 A022168 A022169

Keyword

nonn,tabl

Author

N. J. A. Sloane

Status

approved