A157784 -id:A157784 - cp's OEIS frontend

A022168 Triangle of Gaussian binomial coefficients [ n,k ] for q = 4.

1, 1, 1, 1, 5, 1, 1, 21, 21, 1, 1, 85, 357, 85, 1, 1, 341, 5797, 5797, 341, 1, 1, 1365, 93093, 376805, 93093, 1365, 1, 1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1, 1, 21845, 23859109, 1550842085, 6221613541

Offset: 0

N. J. A. Sloane

The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157784(n,k). - R. J. Mathar, Mar 12 2013

			Triangle begins:
  1;
  1,    1;
  1,    5,       1;
  1,   21,      21,        1;
  1,   85,     357,       85,        1;
  1,  341,    5797,     5797,      341,       1;
  1, 1365,   93093,   376805,    93093,    1365,    1;
  1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1;

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.

T. D. Noe, Rows n=0..50 of triangle, flattened
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint 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.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for sequences related to Gaussian binomial coefficients

Cf. A006118 (row sums), A002450 (k=1), A006105 (k=2), A006106 (k=3).

A022168 := proc(n,m)
        A027637(n)/A027637(n-m)/A027637(m) ;
end proc: # R. J. Mathar, Nov 14 2011

gaussianBinom[n_, k_, q_] := Product[q^i - 1, {i, n}]/Product[q^j - 1, {j, n - k}]/Product[q^l - 1, {l, k}]; Column[Table[gaussianBinom[n, k, 4], {n, 0, 8}, {k, 0, n}], Center] (* Alonso del Arte, Nov 14 2011 *)
Table[QBinomial[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 4; T[n_, 0]:= 1; T[n_,n_]:= 1; T[n_,k_]:= T[n,k] = If[k < 0 || n < k, 0, T[n-1, k -1] +q^k*T[n-1,k]]; Table[T[n,k], {n,0,10}, {k,0,n}] // Flatten  (* G. C. Greubel, May 27 2018 *)

{q=4; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017

G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 4^j - 1. - Seiichi Manyama, May 09 2025

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

cp's OEIS Frontend

A022168 Triangle of Gaussian binomial coefficients [ n,k ] for q = 4.

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