A135950 -id:A135950 - cp's OEIS frontend

A135951 Central terms of triangle A135950, the matrix inverse of triangle A022166.

1, -3, 70, -11160, 12850368, -111842970624, 7558738517524480, -4024873276683363287040, 17013427111087951089139449856, -573105858480900876266937950612226048, 154142404695090288939416498797330749299097600

Offset: 0

Paul D. Hanna, Dec 08 2007

sign

A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.

Cf. A135950, A022166.

max = 20; M = Table[QBinomial[n, k, 2], {n, 0, max}, {k, 0, max}] // Inverse; Table[M[[n, (n+1)/2]], {n, 1, max+1, 2}] (* Jean-François Alcover, Apr 09 2016 *)
Table[(-1)^n 2^((n-1)n/2) QBinomial[2n, n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

a(n)=local(q=2,A=matrix(2*n+1,2*n+1,n,k,if(n>=k,if(n==1 || k==1, 1, prod(j=n-k+1, n-1, 1-q^j)/prod(j=1, k-1, 1-q^j))))^-1);A[2*n+1,n+1]

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

Original entry on oeis.org

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

A158474 Triangle read by rows generated from (x-1)(x-2)(x-4)*...

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -7, 14, -8, 1, -15, 70, -120, 64, 1, -31, 310, -1240, 1984, -1024, 1, -63, 1302, -11160, 41664, -64512, 32768, 1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152, 1, -255, 21590, -777240, 12850368, -99486720, 353730560

Offset: 0

Gary W. Adamson, Mar 20 2009

Row sum of the unsigned triangle = A028361: (1, 2, 6, 30, 270, 4590, ...).
Right border of the unsigned triangle = A006125: (1, 1, 2, 8, 64, 1024, ...).
From Philippe Deléham, Mar 20 2009: (Start)
Unsigned triangle: A077957(n) DELTA A007179(n+1) = [1,0,2,0,4,0,8,0,16,0,32,0,...]DELTA[1,1,4,6,16,28,64,120,256,496,...], where DELTA is the operator defined in A084938.
Signed triangle: [1,0,2,0,4,0,8,0,16,0,...]DELTA[-1,-1,-4,-6,-16,-28,-64,...]. (End)

			First few rows of the triangle =
1;
1,  -1;
1,  -3,     2;
1,  -7,    14,     -8;
1, -15,    70,   -120,       64;
1, -31,   310,  -1240,     1984,    -1024;
1, -63,  1302, -11160,    41664,   -64512,     32768;
1,-127,  5334, -94488,   755904, -2731008,   4161536,  -2097152;
1,-255, 21590,-777240, 12850368,-99486720, 353730560,-534773760, 268435456;
...
Example: row 3 = x^3 - 7x^2 + 14x - 8 = (x-1)*(x-2)*(x-4).

Cf. A028361, A006125.

Cf. A157963, A135950. - R. J. Mathar, Mar 20 2009

A158474 := proc(n,k) mul(x-2^j,j=0..n-1) ; expand(%); coeftayl(%,x=0,n-k) ; end proc: # R. J. Mathar, Aug 27 2011

{{1}}~Join~Table[Reverse@ CoefficientList[Fold[#1 (x - #2) &, 1, 2^Range[0, n]], x], {n, 0, 7}] // Flatten (* Michael De Vlieger, Dec 22 2016 *)

T(n,k) = coefficient [x^(n-k)] of (x-1)*(x-2)*(x-4)*...*(x-2^(n-1)).
T(n,k) = (-1)^k*(Sum_{j=0..k} T(k,j)*2^((k-j)*n))/(Product_{i=1..k} (2^i-1)) for n >= 0 and k > 0, i.e., e.g.f. of col k > 0 is: (-1)^k*(Sum_{j=0..k} T(k,j)* exp(2^(k-j)*t))/(Product_{i=1..k} (2^i-1)). - Werner Schulte, Dec 18 2016
T(n,k)/T(k,k) = A022166(n,k) for 0 <= k <= n. - Werner Schulte, Dec 21 2016

A157783 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (3^(i-1)-x) in row n, column k, 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 3, -4, 1, 27, -39, 13, -1, 729, -1080, 390, -40, 1, 59049, -88209, 32670, -3630, 121, -1, 14348907, -21493836, 8027019, -914760, 33033, -364, 1, 10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093

Offset: 0

Roger L. Bagula, Mar 06 2009

Row sums except n=0 are zero.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=3)=[1,2,9,24,81,234,729,2160,6561,...] DELTA [ -1,0,-3,0,-9,0,-27,0,-81,0,-243,0,...] where DELTA is the operator defined in A084938; see A122006 and A000244. - Philippe Deléham, Mar 09 2009

			Triangle begins
  1;
  1, -1;
  3, -4, 1;
  27, -39, 13, -1;
  729, -1080, 390, -40, 1;
  59049, -88209, 32670, -3630, 121, -1;
  14348907, -21493836, 8027019, -914760, 33033, -364, 1;
  10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093, -1;
  22876792454961, -34309958505840, 12860351387820, -1481851188720, 55340738838, -677572560, 2688780, -3280, 1;
Row n=3 is 27 - 39*x + 13*x^2 - x^3.

Cf. A157832, A135950, A022166, A047656 (column k=1), A003462 (subdiagonal k=n-1), A203243 (subdiagonal k=n-2).

A157783 := proc(n,k)
    product( 3^(i-1)-x,i=1..n) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

Clear[f, q, M, n, m];
q = 3;
f[k_, m_] := If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[k == n && m > 1, 1, 0]]]];
M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
Table[M[n], {n, 1, 10}];
Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
Flatten[a]

A157832 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (5^(i-1)-x) in row n, column k, 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 5, -6, 1, 125, -155, 31, -1, 15625, -19500, 4030, -156, 1, 9765625, -12203125, 2538250, -101530, 781, -1, 30517578125, -38144531250, 7944234375, -319819500, 2542155, -3906, 1, 476837158203125, -596038818359375

Offset: 0

Roger L. Bagula, Mar 07 2009

Except for n=0, the row sums are zero.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=5)= [1,4,25,120,625,3100,15625,...] DELTA [ -1,0,-5,0,-25,0,-125,0,-625,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2009

			Triangle begins
  1;
  1, -1;
  5, -6, 1;
  125, -155, 31, -1;
  15625, -19500, 4030, -156, 1;
  9765625, -12203125, 2538250, -101530, 781, -1;
  30517578125, -38144531250, 7944234375, -319819500, 2542155, -3906, 1;
  476837158203125, -596038818359375, 124166806640625, -5005123921875, 40040991375, -63573405, 19531, -1;

Cf. A135950, A157783, A109345 (first column), A003463 (first subdiagonal).

A157832 := proc(n,k)
        product( 5^(i-1)-x,i=1..n) ;
        coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

p[x_, n_] = If[n == 0, 1, Product[q^(i - 1) - x, {i, 1, n}]];
q = 5;
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

A157784 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (4^(i-1)-x), in row n and column 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 4, -5, 1, 64, -84, 21, -1, 4096, -5440, 1428, -85, 1, 1048576, -1396736, 371008, -23188, 341, -1, 1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1, 4398046511104, -5863704100864, 1563272675328, -99158478848

Offset: 0

Roger L. Bagula, Mar 06 2009

Row sums except n=0 are zero.
The matrix inverses seem to be related to the Gaussian q-form combinations.
Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=4)=[1,3,16,60,256,1008,4096,16320,65536,261888,...] DELTA [ -1,0,-4,0,-16,0,-64,0,-256,0,-1024,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2009

			Triangle begins
  1;
  1, -1;
  4, -5, 1;
  64, -84, 21, -1;
  4096, -5440, 1428, -85, 1;
  1048576, -1396736, 371008, -23188, 341, -1;
  1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1;
  4398046511104, -5863704100864, 1563272675328, -99158478848, 1549351232, -5963412, 5461, -1;
  72057594037927936, -96075326035066880, 25618523216674816, -1626175790120960, 25483729063936, -99253893440, 95436436, -21845, 1;
Row n=3 represents 64 - 84*x + 21*x^2 - x^3.

Cf. A135950, A022166, A157783, A157832.

A157784 := proc(n,k)
    product( 4^(i-1)-x,i=1..n) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

Clear[f, q, M, n, m];
q = 4;
f[k_, m_] := If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[k == n && m > 1, 1, 0]]]];
M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
Table[M[n], {n, 1, 10}];
Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
Flatten[a]

A157785 Triangle of coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.

Original entry on oeis.org

1, 1, -1, -2, 1, 1, -8, 6, 3, -1, 64, -40, -30, 5, 1, 1024, -704, -440, 110, 11, -1, -32768, 21504, 14784, -3080, -462, 21, 1, -2097152, 1409024, 924672, -211904, -26488, 1806, 43, -1, 268435456, -178257920, -119767040, 26199040, 3602368, -204680, -7310, 85, 1

Offset: 0

Roger L. Bagula, Mar 06 2009

Triangle T(n,k), 0 <= k <= n, read by rows given by [1, q-1, q^2, q^3-q, q^4, q^5-q^2, q^6, q^7-q^3, q^8, ...] DELTA [-1, 0, -q, 0, -q^2, 0, -q^3, 0, -q^4, 0, ...] (for q=-2) = [1, -3, 4, -6, 16, -36, 64,...] DELTA [ -1, 0, 2, 0, -4, 0, 8, 0, -16, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2009

			Triangle begins as:
          1;
          1,         -1;
         -2,          1,          1;
         -8,          6,          3,       -1;
         64,        -40,        -30,        5,       1;
       1024,       -704,       -440,      110,      11,      -1;
     -32768,      21504,      14784,    -3080,    -462,      21,     1;
   -2097152,    1409024,     924672,  -211904,  -26488,    1806,    43, -1;
  268435456, -178257920, -119767040, 26199040, 3602368, -204680, -7310, 85, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. this sequence (q=-2), A158020 (q=-1), A007318 (q=1), A157963 (q=2).
Cf. A135950 (q=2; alternative).
Cf. A022166, A135950.

p[x_, n_, q_]:= q^Binomial[n, 2]*QPochhammer[x, 1/q, n];
Table[CoefficientList[Series[p[x, n, -2], {x,0,n}], x], {n,0,10}]//Flatten (* G. C. Greubel, Nov 29 2021 *)

Sum_{k=0..n} T(n, k) = 0^n.
From G. C. Greubel, Nov 29 2021: (Start)
T(n, k) = [x^k] coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.
T(n, k) = [x^k] Product_{j=0..n-1} (q^j - x). (End)

Edited by G. C. Greubel, Nov 29 2021

A157963 Triangle T(n,k), 0<=k<=n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,-q^5,0,...] (for q=2) = [1,1,4,6,16,28,64,...] DELTA [ -1,0,-2,0,-4,0,-8,0,-16,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, -1, 2, -3, 1, 8, -14, 7, -1, 64, -120, 70, -15, 1, 1024, -1984, 1240, -310, 31, -1, 32768, -64512, 41664, -11160, 1302, -63, 1, 2097152, -4161536, 2731008, -755904, 94488, -5334, 127, -1, 268435456, -534773760, 353730560, -99486720, 12850368

Offset: 0

Philippe Deléham, Mar 10 2009

Row sums equal 0^n.
Row n contains the coefficients of Product_{j=0..n-1} (2^j*x-1), highest coefficient first. - Alois P. Heinz, Mar 26 2012
The elements of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^k*A022166(n,k). - R. J. Mathar, Mar 26 2013

			Triangle begins :
1;
1,    -1;
2,    -3,  1;
8,   -14,  7,  -1;
64, -120, 70, -15,  1;

Alois P. Heinz, Rows n = 0..44

Cf. A007179, A130595, A157783, A157784, A157785, A157832, A158474.

T:= n-> seq (coeff (mul (2^j*x-1, j=0..n-1), x, n-k), k=0..n):
seq (T(n), n=0..10);  # Alois P. Heinz, Mar 26 2012

row[n_] := CoefficientList[(-1)^n QPochhammer[x, 2, n] + O[x]^(n+1), x] // Reverse; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 26 2016 *)

T(n,k) = (-1)^n*A135950(n,k). T(n,0) = A006125(n).

T(n,k) = [x^(n-k)] Product_{j=0..n-1} (2^j*x-1). - Alois P. Heinz, Mar 26 2012

A136097 a(n) = A135951(n) /[(2^(n+1)-1) * 2^(n*(n-1)/2)].

Original entry on oeis.org

1, -1, 5, -93, 6477, -1733677, 1816333805, -7526310334829, 124031223014725741, -8152285307423733458541, 2140200604371078953284092525, -2245805993494514875022552272042605, 9423041917569791458584837551185555483245

Offset: 0

Paul D. Hanna, Dec 13 2007

sign

A135951 is the central terms of A135950; A135950 is the matrix inverse of A022166; A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.

Cf. A135951, A135950, A022166.

Table[(-1)^n QBinomial[2n, n, 2]/(2^(n+1) - 1), {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

a(n)=local(q=2,A=matrix(2*n+1,2*n+1,n,k,if(n>=k,if(n==1 || k==1, 1, prod(j=n-k+1, n-1, 1-q^j)/prod(j=1, k-1, 1-q^j))))^-1); A[2*n+1,n+1]/( (q^(n+1)-1)/(q-1) * q^(n*(n-1)/2) )

Conjecture: the n-th central term of the matrix inverse of the triangle of Gaussian binomial coefficients in q is divisible by [(q^(n+1)-1)/(q-1) * q^(n*(n-1)/2)] for n>=0 and integer q > 1.

a(n) = (-1)^n * A015030(n) where A015030 is 2-Catalan numbers. - Michael Somos, Jan 10 2023

A062733 Maximal degree of an irreducible representation of the group SL(n,2) (the group of nonsingular n X n matrices over GF(2) ).

Original entry on oeis.org

1, 2, 8, 70, 1240, 41664, 2731008

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001

			a(2) = 2 because SL(2,2) is isomorphic to the symmetric group S_3 and the degrees are 1,1,2.

Cf. A002884, A135950.

For n > 3, this appears to coincide with the absolute values of the column k=2 of A135950, which would imply a(n) = 2^((n-2)*(n-3)/2) * (2^n - 1) * (2^(n-1) - 1) / 3. - Andrei Zabolotskii, May 27 2025

a(6)-a(7) from Vladeta Jovovic, May 27 2007

cp's OEIS Frontend

A135951 Central terms of triangle A135950, the matrix inverse of triangle A022166.

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

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A158474 Triangle read by rows generated from (x-1)*(x-2)*(x-4)*...

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A157783 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (3^(i-1)-x) in row n, column k, 0 <= k <= n.

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A157832 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (5^(i-1)-x) in row n, column k, 0 <= k <= n.

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A157784 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (4^(i-1)-x), in row n and column 0 <= k <= n.

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A157785 Triangle of coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.

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A158474 Triangle read by rows generated from (x-1)(x-2)(x-4)*...