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
Examples
Triangle begins: 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;
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
Programs
-
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 *)
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
G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 3^j - 1. - Seiichi Manyama, May 09 2025
Comments