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

%I A022167 #73 May 09 2025 08:33:17
%S A022167 1,1,1,1,4,1,1,13,13,1,1,40,130,40,1,1,121,1210,1210,121,1,1,364,
%T A022167 11011,33880,11011,364,1,1,1093,99463,925771,925771,99463,1093,1,1,
%U A022167 3280,896260,25095280,75913222,25095280,896260,3280,1
%N A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3.
%C A022167 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
%D A022167 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%D A022167 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A022167 T. D. Noe, <a href="/A022167/b022167.txt">Rows n = 0..50 of triangle, flattened</a>
%H A022167 R. Mestrovic, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.
%H A022167 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H A022167 M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%H A022167 <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries for sequences related to Gaussian binomial coefficients</a>
%F A022167 T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - _Peter A. Lawrence_, Jul 13 2017
%F A022167 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
%F A022167 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
%e A022167 Triangle begins:
%e A022167   1;
%e A022167   1,    1;
%e A022167   1,    4,      1;
%e A022167   1,   13,     13,        1;
%e A022167   1,   40,    130,       40,        1;
%e A022167   1,  121,   1210,     1210,      121,        1;
%e A022167   1,  364,  11011,    33880,    11011,      364,      1;
%e A022167   1, 1093,  99463,   925771,   925771,    99463,   1093,    1;
%e A022167   1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1;
%p A022167 A022167 := proc(n,m)
%p A022167         A027871(n)/A027871(n-m)/A027871(m) ;
%p A022167 end proc:
%p A022167 seq(seq(A022167(n,m),m=0..n),n=0..10) ; # _R. J. Mathar_, Nov 14 2011
%t A022167 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_ *)
%t A022167 Table[QBinomial[n, k, 3], {n, 0, 10}, {k, 0, n}] // Flatten
%t A022167 (* or, after _Vladimir Kruchinin_, using S for qStirling2: *)
%t A022167 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 A022167 T[n_, k_] /; n >= k := Sum[Binomial[n, j]*S[n-j, n-k, q]*(q-1)^(k-j) /. q -> 3, {j, 0, k}];
%t A022167 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 08 2020 *)
%Y A022167 Columns k=0..3 give A000012, A003462, A006100, A006101.
%Y A022167 Cf. A006117 (row sums).
%K A022167 nonn,tabl
%O A022167 0,5
%A A022167 _N. J. A. Sloane_