The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 2 14:26 EST 2020. Contains 338877 sequences. (Running on oeis4.)