A022171 Triangle of Gaussian binomial coefficients [ n,k ] for q = 7.
1, 1, 1, 1, 8, 1, 1, 57, 57, 1, 1, 400, 2850, 400, 1, 1, 2801, 140050, 140050, 2801, 1, 1, 19608, 6865251, 48177200, 6865251, 19608, 1, 1, 137257, 336416907, 16531644851, 16531644851, 336416907, 137257, 1, 1
Offset: 0
Examples
Triangle begins: 1; 1, 1; 1, 8, 1; 1, 57, 57, 1; 1, 400, 2850, 400, 1; 1, 2801, 140050, 140050, 2801, 1; 1, 19608, 6865251, 48177200, 6865251, 19608, 1; 1, 137257, 336416907, 16531644851, 16531644851, 336416907, 137257, 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: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
-
Maple
A027875 := proc(n) mul(7^i-1,i=1..n) ; end proc: A022171 := proc(n,m) A027875(n)/A027875(m)/A027875(n-m) ; end proc: # R. J. Mathar, Jul 19 2017
-
Mathematica
p[n_]:=Product[7^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,7], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 7; 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=7; 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) = 7^j - 1. - Seiichi Manyama, May 09 2025