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 A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3. 22
 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 (list; table; graph; refs; listen; history; text; internal format)
 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) 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 * A064281 A267318 A050154 Adjacent sequences:  A022164 A022165 A022166 * A022168 A022169 A022170 KEYWORD nonn,tabl AUTHOR STATUS approved

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Last modified October 24 04:47 EDT 2020. Contains 337975 sequences. (Running on oeis4.)