A022168 Triangle of Gaussian binomial coefficients [ n,k ] for q = 4.
1, 1, 1, 1, 5, 1, 1, 21, 21, 1, 1, 85, 357, 85, 1, 1, 341, 5797, 5797, 341, 1, 1, 1365, 93093, 376805, 93093, 1365, 1, 1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1, 1, 21845, 23859109, 1550842085, 6221613541
Offset: 0
Examples
Triangle begins: 1; 1, 1; 1, 5, 1; 1, 21, 21, 1; 1, 85, 357, 85, 1; 1, 341, 5797, 5797, 341, 1; 1, 1365, 93093, 376805, 93093, 1365, 1; 1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1;
References
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
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
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Maple
A022168 := proc(n,m) A027637(n)/A027637(n-m)/A027637(m) ; end proc: # R. J. Mathar, Nov 14 2011
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Mathematica
gaussianBinom[n_, k_, q_] := Product[q^i - 1, {i, n}]/Product[q^j - 1, {j, n - k}]/Product[q^l - 1, {l, k}]; Column[Table[gaussianBinom[n, k, 4], {n, 0, 8}, {k, 0, n}], Center] (* Alonso del Arte, Nov 14 2011 *) Table[QBinomial[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 4; T[n_, 0]:= 1; T[n_,n_]:= 1; T[n_,k_]:= T[n,k] = If[k < 0 || n < k, 0, T[n-1, k -1] +q^k*T[n-1,k]]; Table[T[n,k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, May 27 2018 *) -
PARI
{q=4; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || n
Formula
T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017
G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 4^j - 1. - Seiichi Manyama, May 09 2025
Comments