A022166 -id:A022166 - cp's OEIS frontend

A290974 Alternating sum of row 2n of A022166.

1, -1, 7, -217, 27559, -14082649, 28827182503, -236123451882073, 7737057147819885991, -1014103817421900276726361, 531681448124675830384033629607, -1115016280616112042365706510363949657, 9353433376690281791373262192784600640357799

Offset: 0

Geoffrey Critzer, Aug 16 2017

sign

The alternating row sums of A022166(n,k) is zero when n is odd.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
A. Nijenhuis, A. E. Solow, and H. S. Wilf, Bijective Methods in the Theory of Finite Vector Spaces, Journal of Combinatorial Theory, Series A 37,(1984), 80-84.

Cf. A005329, A022166.

nn = 26; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}]; Select[Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}] CoefficientList[Series[eq[-z]*eq[z] /. q -> 2, {z, 0, nn}], z], # != 0 &]
a[n_Integer] := a[n] = 2 QPochhammer[1/2, 4, n + 1];
Table[a[n], {n, 0, 12}] (* Vladimir Reshetnikov, Sep 23 2021 *)

a(n) = Sum_{k=0..2n} (-1)^k A022166(2n,k).
a(0) = 1, a(n) = (1 - 2^(2n-1))*a(n-1).
a(n)/A005329(2n) is the coefficient of z^(2n) in the expansion of eq(-z)*eq(z) where eq(z) is the q-exponential function.
O.g.f.: Sum_{n>=0} a(n)*x^n = 1/(1 + (q-1)*x/(1 + q*(q^2-1)*x/(1 + q^2*(q^3-1)*x/(1 + q^3*(q^4-1)*x/(1 + q^4*(q^5-1)*x/(1 + q^5*(q^6-1)*x/(1 + ...))))))), a continued fraction, when evaluated at q = 2. - Paul D. Hanna, Aug 29 2020
O.g.f.: Sum_{n>=0} a(n)*x^(2*n) = Sum_{n>=0} (-x)^k / Product{k=0..n} (1 - 2^k*x). - Paul D. Hanna, Aug 29 2020

A001576 a(n) = 1^n + 2^n + 4^n.

Original entry on oeis.org

3, 7, 21, 73, 273, 1057, 4161, 16513, 65793, 262657, 1049601, 4196353, 16781313, 67117057, 268451841, 1073774593, 4295032833, 17180000257, 68719738881, 274878431233, 1099512676353, 4398048608257, 17592190238721, 70368752566273, 281474993487873, 1125899940397057

Offset: 0

N. J. A. Sloane

Equals A135576, except for the first term. - Omar E. Pol, Nov 18 2008
Conjecture: For n > 1, if a(n) = 1^n + 2^n + 4^n is a prime number then n is of the form 3^h. For example, for h=1, n=3, a(n) = 1^3 + 2^3 + 4^3 = 73 (prime); for h=2, n=9, a(n) = 1^9 + 2^9 + 4^9 = 262657 (prime); for h=3, n=27, a(n) is not prime. - Vincenzo Librandi, Aug 03 2010
The previous conjecture was proved by Golomb in 1978. See A051154. - T. D. Noe, Aug 15 2010
Another more elementary proof can be found in Liu link. - Bernard Schott, Mar 08 2019
Fills in one quarter section of the figurate form of the Sierpinski square curve. See illustration in links and A141725. - John Elias, Mar 29 2023

T. D. Noe, Table of n, a(n) for n = 0..200
John Elias, Illustration of Initial Terms: 1/4 Sierpinski Square Curve
Andy Liu, West German Mathematical Olympiad 1982 - Second round, Problem 4, Crux Mathematicorum, p. 105, Vol. 12, May. 86.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Subsequence of A002061.
Cf. A001550, A034513, A001579, A074501-A074580, A135576, A135577.
See also comments in A051154.
Cf. A006095, A022166.

Mathematica
```
Table[1^n + 2^n + 4^n, {n, 0, 24}]
```

a(n)=1+2^n+4^n \\ Charles R Greathouse IV, Jun 10 2011

[sigma(4,n)for n in range(0,23)] # Zerinvary Lajos, Jun 04 2009

a(n) = 6*a(n-1) - 8*a(n-2) + 3.
O.g.f.: -1/(-1+x) - 1/(-1+2*x) - 1/(-1+4*x) = ( -3+14*x-14*x^2 ) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Feb 29 2008
E.g.f.: e^x + e^(2*x) + e^(4*x). - Mohammad K. Azarian, Dec 26 2008
a(n) = A024088(n)/A000225(n). - Reinhard Zumkeller, Feb 15 2009
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 7*x + 35*x^2 + 155*x^3 + ... is the o.g.f. for the 2nd subdiagonal of triangle A022166, essentially A006095. - Peter Bala, Apr 07 2015

A015109 Triangle of Gaussian (or q-binomial) coefficients for q = -2.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 3, 3, 1, 1, -5, 15, -5, 1, 1, 11, 55, 55, 11, 1, 1, -21, 231, -385, 231, -21, 1, 1, 43, 903, 3311, 3311, 903, 43, 1, 1, -85, 3655, -25585, 56287, -25585, 3655, -85, 1, 1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1, 1, -341, 58311

Offset: 0

Olivier Gérard

May be read as a symmetric triangular (T(n,k)=T(n,n-k); k=0,...,n; n=0,1,...) or square array (A(n,r)=A(r,n)=T(n+r,r), read by antidiagonals). The diagonals of the former (or rows/columns of the latter) are A000012 (k=0), A077925 (k=1), A015249 (k=2), A015266 (k=3), A015287 (k=4), A015305 (k=5), A015323 (k=6), A015338 (k=7), A015356 (k=8), A015371 (k=9), A015386 (k=10), A015405 (k=11), A015423 (k=12), ... - M. F. Hasler, Nov 04 2012
The elements of the inverse matrix are apparently T^(-1)(n,k) = (-1)^n*A157785(n,k). - R. J. Mathar, Mar 12 2013
Fu et al. give two combinatorial interpretations of the (unsigned) q-binomial coefficients when q is a negative integer. - Peter Bala, Nov 02 2017

			From _Roger L. Bagula_, Feb 10 2009: (Start)
  1;
  1,   1;
  1,  -1,     1;
  1,   3,     3,      1;
  1,  -5,    15,     -5,      1;
  1,  11,    55,     55,     11,      1;
  1, -21,   231,   -385,    231,    -21,      1;
  1,  43,   903,   3311,   3311,    903,     43,     1;
  1, -85,  3655, -25585,  56287, -25585,   3655,   -85,   1;
  1, 171, 14535, 208335, 875007, 875007, 208335, 14535, 171, 1;  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations of Fractional Supersymmetry, arxiv:hep-th/9506177 (1995).
S. Fu, V. Reiner, D. Stanton and N. Thiem, The negative q-binomial, arXiv:1108.4702 [math.CO], 2011.
R. Parthasarathy, q-Fermionic Numbers and Their Roles in Some Physical Problems, arxiv:quant-ph/0403216, 2004.

Cf. A015152 (row sums).
Cf. A022166 (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: 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).

qBinomial:= func< n,k,q | k eq 0 select 1 else (&*[(1-q^(n-j+1))/(1-q^j): j in [1..k]]) >;
[qBinomial(n,k,-2): k in [0..n], n in [0..10]]; // A015109 // G. C. Greubel, Nov 30 2021

A015109 := proc(n, k)
   mul( ((-2)^(1+n-i)-1)/((-2)^i-1) ,i=1..k) ;
end proc: # R. J. Mathar, Mar 12 2013

T[n_, k_, q_]:= Product[(1 - q^(n-j+1))/(1 - q^j), {j, k}];
Table[T[n,k,-2], {n,0,10}, {k,0,n}]//Flatten (* Roger L. Bagula, Feb 10 2009 *)(* modified by G. C. Greubel, Nov 30 2021 *)
Table[QBinomial[n, k, -2], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, Apr 09 2016 *)

T015109(n, k, q=-2)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) \\ M. F. Hasler, Nov 04 2012

flatten([[q_binomial(n,k,-2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Nov 30 2021

T(n, k) = q-binomial(n, k, -2).

T(n, k, q) = Product_{j=1..k} ( (1 - q^(n-j+1))/(1 - q^j) ), for q = -2. - Roger L. Bagula, Feb 10 2009

Edited by M. F. Hasler, Nov 04 2012

A006116 Sum of Gaussian binomial coefficients [n,k] for q=2 and k=0..n.

Original entry on oeis.org

1, 2, 5, 16, 67, 374, 2825, 29212, 417199, 8283458, 229755605, 8933488744, 488176700923, 37558989808526, 4073773336877345, 623476476706836148, 134732283882873635911, 41128995468748254231002, 17741753171749626840952685, 10817161765507572862559462656

Offset: 0

N. J. A. Sloane

Also number of distinct binary linear codes of length n and any dimension.
Equivalently, number of subgroups of the Abelian group (C_2)^n.
Let V_n be an n-dimensional vector space over a field with 2 elements. Let P(V_n) be the collection of all subspaces of V_n. Then a(n-1) is the number of times any given nonzero vector of V_n appears in P(V_n). - Geoffrey Critzer, Jun 05 2017
With V_n and P(V_n) as above, a(n) is also the cardinality of P(V_n). - Vaia Patta, Jun 25 2019

			O.g.f.: A(x) = 1/(1-x) + x/((1-x)*(1-2x)) + x^2/((1-x)*(1-2x)*(1-4x)) + x^3/((1-x)*(1-2x)*(1-4x)*(1-8x)) + ...
Also generated by iterated binomial transforms in the following way:
[1,2,5,16,67,374,2825,29212,...] = BINOMIAL([1,1,2,6,26,158,1330,...]); see A135922;
[1,2,6,26,158,1330,15414,245578,...] = BINOMIAL([1,1,3,13,83,749,...]);
[1,3,13,83,749,9363,160877,...] = BINOMIAL^2([1,1,5,33,317,4361,...]);
[1,5,33,317,4361,82789,2148561,...] = BINOMIAL^4([1,1,9,97,1433,...]);
[1,9,97,1433,30545,902601,...] = BINOMIAL^8([1,1,17,321,7601,252833,...]);
etc.

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.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
S. Hitzemann and W. Hochstattler, On the combinatorics of Galois numbers, Discr. Math. 310 (2010) 3551-3557, Galois Numbers G_{n}^(2).
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.
Vjekoslav Kovač and Hrvoje Šikić, Characterizations of democratic systems of translates on locally compact abelian groups, arXiv:1709.01747 [math.FA], 2017.
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)
Index entries for sequences related to binary linear codes

Cf. A006516. Row sums of A022166.

Cf. A005329, A083906. - Paul D. Hanna, Nov 29 2008

I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1)+(2^(n-2)-1)*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Aug 12 2014

gf:= m-> add(x^n/mul(1-2^k*x, k=0..n), n=0..m):
a:= n-> coeff(series(gf(n), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 24 2012
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 1,
      2^m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2021

faq[n_, q_] = Product[(1-q^(1+k))/(1-q), {k, 0, n-1}]; qbin[n_, m_, q_] = faq[n, q]/(faq[m, q]*faq[n-m, q]); a[n_] := Sum[qbin[n, k, 2], {k, 0, n}]; a /@ Range[0, 19] (* Jean-François Alcover, Jul 21 2011 *)
Flatten[{1, RecurrenceTable[{a[n]==2*a[n-1]+(2^(n-1)-1)*a[n-2], a[0]==1, a[1]==2}, a, {n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
QP = QPochhammer; a[n_] := Sum[QP[2, 2, n]/(QP[2, 2, k]*QP[2, 2, n-k]), {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 23 2015 *)
Table[Sum[QBinomial[n, k, 2], {k, 0, n}], {n, 0, 19}] (* Ivan Neretin, Mar 28 2016 *)

a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-2^j*x+x*O(x^n))), n) \\ Paul D. Hanna, Dec 06 2007

a(n,q=2)=sum(k=0,n,prod(i=1,n-k,(q^(i+k)-1)/(q^i-1))) \\ Paul D. Hanna, Nov 29 2008

O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - 2^k*x). - Paul D. Hanna, Dec 06 2007
From Paul D. Hanna, Nov 29 2008: (Start)
Coefficients of the square of the q-exponential of x evaluated at q=2, where the q-exponential of x = Sum_{n>=0} x^n/F(n) and F(n) = Product{i=1..n} (q^i-1)/(q-1) is the q-factorial of n.
G.f.: (Sum_{k=0..n} x^n/F(n))^2 = Sum_{k=0..n} a(n)*x^n/F(n) where F(n) = A005329(n) = Product{i=1..n} (2^i - 1).
a(n) = Sum_{k=0..n} F(n)/(F(k)*F(n-k)) where F(n)=A005329(n) is the 2-factorial of n.
a(n) = Sum_{k=0..n} Product_{i=1..n-k} (2^(i+k) - 1)/(2^i - 1).
a(n) = Sum_{k=0..A033638(n)} A083906(n,k)*2^k. (End)
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-2^k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
a(n) = 2*a(n-1) + (2^(n-1)-1)*a(n-2). [Hitzemann and Hochstattler]. - R. J. Mathar, Aug 21 2013
a(n) ~ c * 2^(n^2/4), where c = EllipticTheta[3,0,1/2] / QPochhammer[1/2,1/2] = 7.3719688014613... if n is even and c = EllipticTheta[2,0,1/2] / QPochhammer[1/2,1/2] = 7.3719494907662... if n is odd. - Vaclav Kotesovec, Aug 21 2013

A015129 Triangle of (Gaussian) q-binomial coefficients for q = -13.

Original entry on oeis.org

1, 1, 1, 1, -12, 1, 1, 157, 157, 1, 1, -2040, 26690, -2040, 1, 1, 26521, 4508570, 4508570, 26521, 1, 1, -344772, 761974851, -9900819720, 761974851, -344772, 1, 1, 4482037, 128773405047, 21752862899691, 21752862899691, 128773405047, 4482037, 1

Offset: 0

Olivier Gérard

May be read as a symmetric triangular (T(n,k) = T(n,n-k); k=0,...,n; n=0,1,...) or square array (A(n,r) = A(r,n) = T(n+r,r), read by antidiagonals). The diagonals of the former, resp. rows (or columns) of the latter, are: A000012 (all 1's), A015000 (q-integers for q=-13), A015265 (k=2), A015286 (k=3), A015303 (k=4), A015321 (k=5), A015337 (k=6), A015355 (k=7), A015370 (k=8), A015385 (k=9), A015402 (k=10), A015422 (k=11), A015438 (k=12). - M. F. Hasler, Nov 04 2012

			The square array looks as follows:
1    1          1              1                      1               1       ...
1   -12        157           -2040                  26521          -344772    ...
1   157       26690         4508570               761974851      128773405047 ...
1  -2040     4508570      -9900819720           21752862899691        ...
1  26521    761974851    21752862899691       621305270140974342      ...
1 -344772 128773405047 -47790911017216080  17745052029585350965782    ...
(...)

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for other negative q=-2,...,-15: 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), A015132 (q=-14), A015133 (q=-15). - M. F. Hasler, Nov 04 2012

Cf. analog triangles for positive q=2,...,24: A022166 (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). - M. F. Hasler, Nov 05 2012

qBinomial:= func< n,k,q | k eq 0 select 1 else (&*[(1 -q^(n-j+1))/(1 -q^j): j in [1..k]]) >;
[qBinomial(n,k,-13): k in [0..n], n in [0..10]]; // A015129 // G. C. Greubel, Dec 01 2021

Flatten[Table[QBinomial[x,y,-13],{x,0,10},{y,0,x}]] (* Harvey P. Dale, Jul 12 2014 *)

A015129(n, r, q=-13)=prod(i=1, r, (q^(1+n-i+r)-1)/(q^i-1)) \\ (Indexing is that of the square array: n,r=0,1,2,...) - M. F. Hasler, Nov 03 2012

flatten([[q_binomial(n,k,-13) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Dec 01 2021

As a triangle, T(n, k) = Product_{i=1..k} ((-13)^(1+n-i)-1)/((-13)^i-1), with 0 <= k <= n = 0,1,2,...

A076831 Triangle T(n,k) read by rows giving number of inequivalent binary linear [n,k] codes (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 16, 22, 16, 6, 1, 1, 7, 23, 43, 43, 23, 7, 1, 1, 8, 32, 77, 106, 77, 32, 8, 1, 1, 9, 43, 131, 240, 240, 131, 43, 9, 1, 1, 10, 56, 213, 516, 705, 516, 213, 56, 10, 1, 1, 11, 71, 333, 1060, 1988, 1988

Offset: 0

N. J. A. Sloane, Nov 21 2002

"The familiar appearance of the first few rows [...] provides a good example of the perils of too hasty extrapolation in mathematics." - Slepian.

The difference between this triangle and the one for which it can be so easily mistaken is A250002. - Tilman Piesk, Nov 10 2014.

			     k    0   1   2   3    4    5    6    7    8   9  10  11        sum
   n
   0      1                                                           1
   1      1   1                                                       2
   2      1   2   1                                                   4
   3      1   3   3   1                                               8
   4      1   4   6   4    1                                         16
   5      1   5  10  10    5    1                                    32
   6      1   6  16  22   16    6    1                               68
   7      1   7  23  43   43   23    7    1                         148
   8      1   8  32  77  106   77   32    8    1                    342
   9      1   9  43 131  240  240  131   43    9   1                848
  10      1  10  56 213  516  705  516  213   56  10   1           2297
  11      1  11  71 333 1060 1988 1988 1060  333  71  11   1       6928

M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint No. 1693, Tech. Hochschule Darmstadt, 1994

Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [This is a rectangular array whose lower triangle contains T(n,k).]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Apparently, the notation for T(n,k) is W_{nk2}; see p. 197.]
Petros Hadjicostas, Generating function for column k=4.
Petros Hadjicostas, Generating function for column k=5.
Petros Hadjicostas, Generating function for column k=6.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
Marcel Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
Marcel Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
Marcel Wild, The asymptotic number of binary codes and binary matroids, SIAM J. Discrete Math. 19 (2005), 691-699. [This paper apparently corrects some errors in previous papers.]
Index entries for sequences related to binary linear codes

Cf. A006116, A022166, A076766 (row sums).
A034356 gives same table but with the k=0 column omitted.
Columns include A000012 (k=0), A000027 (k=1), A034198 (k=2), A034357 (k=3), A034358 (k=4), A034359 (k=5), A034360 (k=6), A034361 (k=7), A034362 (k=8).

# Fripertinger's method to find the g.f. of column k >= 2 (for small k):
def A076831col(k, length):
    G1 = PSL(k, GF(2))
    G2 = PSL(k-1, GF(2))
    D1 = G1.cycle_index()
    D2 = G2.cycle_index()
    f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
    f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
    f = (f1 - f2)/(1-x)
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 gives
print(A076831col(4, 30)) # Petros Hadjicostas, Sep 30 2019

From Petros Hadjicostas, Sep 30 2019: (Start)
T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n.
G.f. for column k=2: -(x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^4).
G.f. for column k=3: -(x^12 - 2*x^11 + x^10 - x^9 - x^6 + x^4 - x - 1)*x^3/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^2 + x + 1)^2*(x^2 + 1)*(x + 1)^2*(x - 1)^8).
G.f. for column k >= 4: modify the Sage program below (cf. function f). It is too complicated to write it here. (See also some of the links above.)
(End)

A015110 Triangle of q-binomial coefficients for q=-3.

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, 7, 7, 1, 1, -20, 70, -20, 1, 1, 61, 610, 610, 61, 1, 1, -182, 5551, -15860, 5551, -182, 1, 1, 547, 49777, 433771, 433771, 49777, 547, 1, 1, -1640, 448540, -11662040, 35569222, -11662040, 448540, -1640, 1, 1, 4921, 4035220, 315323620

Offset: 0

Olivier Gérard

May be read as a symmetric triangular (T[n,k]=T[n,n-k]; k=0,...,n; n=0,1,...) or square array (A[n,r]=A[r,n]=T[n+r,r], read by antidiagonals). The diagonals of the former (or rows/columns of the latter) are A000012 (k=0), A014983 (k=1), A015251 (k=2), A015268 (k=3), A015288 (k=4), A015306 (k=5), A015324 (k=6), A015340 (k=7), A015357 (k=8), A015375 (k=9), A015388 (k=10), A015407 (k=11), A015424 (k=12),... - M. F. Hasler, Nov 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. analog triangles for other q: A015109 (q=-2), 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); A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 04 2012

Flatten[Table[QBinomial[n, m, -3], {n, 0, 50}, {m, 0, n}]] (* Vincenzo Librandi, Nov 01 2012 *)

T015110(n, k, q=-3)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012

A015117 Triangle of q-binomial coefficients for q=-7.

Original entry on oeis.org

1, 1, 1, 1, -6, 1, 1, 43, 43, 1, 1, -300, 2150, -300, 1, 1, 2101, 105050, 105050, 2101, 1, 1, -14706, 5149551, -35927100, 5149551, -14706, 1, 1, 102943, 252313293, 12328144851, 12328144851, 252313293, 102943, 1, 1, -720600, 12363454300

Offset: 0

Olivier Gérard

May be read as a symmetric triangular (T[n,k]=T[n,n-k]; k=0,...,n; n=0,1,...) or square array (A[n,r]=A[r,n]=T[n+r,r], read by antidiagonals). The diagonals of the former (or rows/columns of the latter) are A000012 (k=0), A014989 (k=1), A015258 (k=2), A015275, A015293, A015312, A015330, A015346, A015363, A015379, A015393 (k=10), A015411, A015430,... - M. F. Hasler, Nov 04 2012

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for negative q=-2,...,-15: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15);

analog triangles for positive q=2,...,24: A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 04 2012

Flatten[Table[QBinomial[n,m,-7],{n,0,10},{m,0,n}]] (* Harvey P. Dale, Aug 08 2012 *)

T015117(n, k, q=-7)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012

A015133 Triangle of (Gaussian) q-binomial coefficients for q=-15.

Original entry on oeis.org

1, 1, 1, 1, -14, 1, 1, 211, 211, 1, 1, -3164, 47686, -3164, 1, 1, 47461, 10726186, 10726186, 47461, 1, 1, -711914, 2413439311, -36190151564, 2413439311, -711914, 1, 1, 10678711, 543023133061, 122144174967811, 122144174967811, 543023133061

Offset: 0

Olivier Gérard

Index entries related to Gaussian binomial coefficients.

Cf. analog triangles for other negative q=-2,...,-15: 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). - M. F. Hasler, Nov 04 2012

Cf. analog triangles for positive q=2,...,24: A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 05 2012

A015133(n, r, q=-13)=prod(i=1, r, (q^(1+n-i+r)-1)/(q^i-1)) \\ (Indexing is that of the square array: n,r=0,1,2,...) - M. F. Hasler, Nov 03 2012

As a triangle, T(n, k) = Product_{i=1..k} ((-15)^(n-i+1)-1)/((-15)^i-1), with 0 <= k <= n = 0,1,2,... - M. F. Hasler, Nov 05 2012

A015112 Triangle of q-binomial coefficients for q=-4.

Original entry on oeis.org

1, 1, 1, 1, -3, 1, 1, 13, 13, 1, 1, -51, 221, -51, 1, 1, 205, 3485, 3485, 205, 1, 1, -819, 55965, -219555, 55965, -819, 1, 1, 3277, 894621, 14107485, 14107485, 894621, 3277, 1, 1, -13107, 14317213, -901984419, 3625623645, -901984419, 14317213, -13107, 1, 1

Offset: 0

Olivier Gérard

May be read as a symmetric triangular (T[n,k]=T[n,n-k]; k=0,...,n; n=0,1,...) or square array (A[n,r]=A[r,n]=T[n+r,r], read by antidiagonals). The diagonals of the former (or rows/columns of the latter) are A000012 (k=0), A014985 (k=1), A015253 (k=2), A015271, A015289, A015308, A015326, A015341, A015359, A015376, A015390 (k=10), A015408, A015425,... - M. F. Hasler, Nov 04 2012

Cf. analog triangles for other q: A015109 (q=-2), A015110 (q=-3), 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); A022166 (q=2), A022167 (q=3), A022168, A022169, A022170, A022171, A022172, A022173, A022174 (q=10), A022175, A022176, A022177, A022178, A022179, A022180, A022181, A022182, A022183, A022184 (q=20), A022185, A022186, A022187, A022188. - M. F. Hasler, Nov 04 2012

Flatten[Table[QBinomial[n,m,-4],{n,0,10},{m,0,n}]] (* Harvey P. Dale, Jun 10 2015 *)

T015112(n, k, q=-4)=prod(i=1, k, (q^(1+n-i)-1)/(q^i-1)) \\ (Indexing is that of the triangular array: 0 <= k <= n = 0,1,2,...) - M. F. Hasler, Nov 04 2012

cp's OEIS Frontend

A290974 Alternating sum of row 2n of A022166.

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A001576 a(n) = 1^n + 2^n + 4^n.

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A015109 Triangle of Gaussian (or q-binomial) coefficients for q = -2.

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A006116 Sum of Gaussian binomial coefficients [n,k] for q=2 and k=0..n.

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A015129 Triangle of (Gaussian) q-binomial coefficients for q = -13.

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A076831 Triangle T(n,k) read by rows giving number of inequivalent binary linear [n,k] codes (n >= 0, 0 <= k <= n).

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