A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3.

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 130, 40, 1, 1, 121, 1210, 1210, 121, 1, 1, 364, 11011, 33880, 11011, 364, 1, 1, 1093, 99463, 925771, 925771, 99463, 1093, 1, 1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1

Offset

0,5

Comments

The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157783(n,k). - R. J. Mathar, Mar 12 2013

References

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

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

Formula

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

T(n,k) = Sum_{j=0..k} C(n,j)*qStirling2(n-j,n-k,3)*(2)^(k-j),j,0,k), n >= k, where qStirling2(n,k,3) is triangle A333143. - Vladimir Kruchinin, Mar 07 2020

Example

1;
1, 1;
1, 4, 1;
1, 13, 13, 1;
1, 40, 130, 40, 1;
1, 121, 1210, 1210, 121, 1;
1, 364, 11011, 33880, 11011, 364, 1;
1, 1093, 99463, 925771, 925771, 99463, 1093, 1;
1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1;

Maple

A022167 := proc(n, m)
        A027871(n)/A027871(n-m)/A027871(m) ;
end proc:
seq(seq(A022167(n, m), m=0..n), n=0..10) ; # R. J. Mathar, Nov 14 2011

Mathematica

p[n_] := Product[3^k-1, {k, 1, n}]; t[n_, m_] := p[n]/(p[n-m]*p[m]); Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014, after R. J. Mathar *)
Table[QBinomial[n, k, 3], {n, 0, 10}, {k, 0, n}] // Flatten
(* or, after Vladimir Kruchinin, using S 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 -> 3, {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 08 2020 *)

Crossrefs

Cf. A006117 (row sums), A003462 (column k=1), A006100 (k=2), A006101 (k=3).

Sequence in context: A157153 A212801 A147565 * A339849 A064281 A267318

Adjacent sequences: A022164 A022165 A022166 * A022168 A022169 A022170

Keyword

nonn,tabl

Author

N. J. A. Sloane

Status

approved