A006098 -id:A006098 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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;

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.

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

Cf. A006516, A218449, A135950 (matrix inverse), A000225 (k=1), A006095 (k=2), A006096 (k=3), A139382.
Cf. this sequence (q=2), A022167 (q=3), A022168 (q=4), A022169 (q=5), A022170 (q=6), A022171 (q=7), A022172 (q=8), A022173 (q=9), A022174 (q=10), A022175 (q=11), A022176 (q=12), A022177 (q=13), A022178 (q=14), A022179 (q=15), A022180 (q=16), A022181 (q=17), A022182 (q=18), A022183 (q=19), A022184 (q=20), A022185 (q=21), A022186 (q=22), A022187 (q=23), A022188 (q=24).
Analogous triangles for other q: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15).

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

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

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 *)

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

qp = matpascal(9,2);
for(n=1,#qp,for(k=1,n,print1(qp[n,k],", "))) \\ Gerald McGarvey, Dec 05 2009

{q=2; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

def T(n,k): return gaussian_binomial(n,k).subs(q=2) # Ralf Stephan, Mar 02 2014

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

A218449 Gaussian binomial coefficient [2*n-1,n] for q=2, n>=0.

Original entry on oeis.org

1, 1, 7, 155, 11811, 3309747, 3548836819, 14877590196755, 246614610741341843, 16256896431763117598611, 4274137206973266943778085267, 4488323837657412597958687922896275, 18839183877670041942218307147122500601235

Offset: 0

Paul D. Hanna, Oct 28 2012

nonn

Compare to: [x^n] Product_{k=0..n-1} 1+2^k*x = 2^(n*(n-1)/2).

			The coefficients in Product_{k=0..n-1} 1/(1 - 2^k*x) begin:
n=0: [(1)];
n=1: [1,(1), 1, 1, 1, 1, 1, 1, 1, 1, ...];
n=2: [1, 3,(7), 15, 31, 63, 127, 255, 511, 1023, ...];
n=3: [1, 7, 35,(155), 651, 2667, 10795, 43435, 174251, ...];
n=4: [1, 15, 155, 1395,(11811), 97155, 788035, 6347715, ...];
n=5: [1, 31, 651, 11811, 200787,(3309747), 53743987, ...];
n=6: [1, 63, 2667, 97155, 3309747, 109221651,(3548836819), ...];
n=7: [1, 127, 10795, 788035, 53743987, 3548836819, 230674393235,(14877590196755), ...]; ...
the coefficients in parenthesis give the initial terms of this sequence;
an adjacent diagonal forms the Gaussian binomial coefficients [2*n,n] for q=2:
[1, 3, 35, 1395, 200787, 109221651, 230674393235, ...] = A006098.

G. C. Greubel, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A022166, A006098, A006125.

Table[QBinomial[2n-1, n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 12 2016 *)

{a(n)=polcoeff(prod(k=0,n-1,1/(1-2^k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

a(n) = [x^n] Product_{k=0..n-1} 1/(1 - 2^k*x).

a(n) ~ c * 2^(n*(n-1)), where c = A065446. - Vaclav Kotesovec, Sep 22 2016

A384038 Number of 2n X 2n matrices M over GF(2) such that the column space of M is equal to the null space of M.

Original entry on oeis.org

1, 3, 210, 234360, 4047865920, 1092146608143360, 4650098142288472473600, 314462403262051153026062745600, 338960040818652280796119613717033779200, 5834618256563872511581456247120956565738854809600, 1605370810586153268821245248112723240374305354675084328960000

Offset: 0

Geoffrey Critzer, May 17 2025

nonn

Let M be a 2n X 2n matrix over GF(2) such that the column space of M is equal to the null space of M. Then M is idempotent and nullity(M) = n and index(M) = 2. If M' is similar to M then the column space of M' equals the null space of M'. Moreover, all such matrices are in the same similarity class (see Hoffman link).

			a(1) = 3 because there are 3 matrices of size 2 X 2 over GF(2) with the desired property: {{0, 0}, {1, 0}}, {{0, 1}, {0, 0}}, {{1, 1}, {1, 1}}.

D. G. Hoffman, Digraphs of finite linear transformations, Australasion Journal of Combinatorics,12: 225-238 (1995).

Cf. A002884, A006098, A346214, A053763.

q = 2; b[p_, i_] := Count[p, i]; d[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}]; aut[deg_, p_] := Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; Table[Product[2^(2 k) - 2^i, {i, 0, (2 k) - 1}]/aut[1, Table[2, {k}]], {k,0, 10}]

a(n) = A002884(n)*A006098(n).

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

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A218449 Gaussian binomial coefficient [2*n-1,n] for q=2, n>=0.

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A384038 Number of 2n X 2n matrices M over GF(2) such that the column space of M is equal to the null space of M.

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