A022170 Triangle of Gaussian binomial coefficients [ n,k ] for q = 6.
1, 1, 1, 1, 7, 1, 1, 43, 43, 1, 1, 259, 1591, 259, 1, 1, 1555, 57535, 57535, 1555, 1, 1, 9331, 2072815, 12485095, 2072815, 9331, 1, 1, 55987, 74630671, 2698853335, 2698853335, 74630671, 55987, 1, 1, 335923
Offset: 0
Examples
Triangle begins: 1; 1, 1; 1, 7, 1; 1, 43, 43, 1; 1, 259, 1591, 259, 1; 1, 1555, 57535, 57535, 1555, 1; 1, 9331, 2072815, 12485095, 2072815, 9331, 1; 1, 55987, 74630671, 2698853335, 2698853335, 74630671, 55987, 1 ;
References
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
Links
- G. C. Greubel, 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.
- Index entries for sequences related to Gaussian binomial coefficients
Programs
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Maple
A027873 := proc(n) mul(6^i-1,i=1..n) ; end procc: A022170 := proc(n,m) A027873(n)/A027873(m)/A027873(n-m) ; end proc: # R. J. Mathar, Jul 19 2017
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Mathematica
p[n_]:= Product[6^i - 1, {i, 1, n}]; t[n_, k_]:= p[n]/(p[k]*p[n-k]); Table[t[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* Vincenzo Librandi, Aug 13 2016 *) Table[QBinomial[n,k,6], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 6; 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=6; 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) = 6^j - 1. - Seiichi Manyama, May 09 2025