|
|
A022169
|
|
Triangle of Gaussian binomial coefficients [ n,k ] for q = 5.
|
|
20
|
|
|
1, 1, 1, 1, 6, 1, 1, 31, 31, 1, 1, 156, 806, 156, 1, 1, 781, 20306, 20306, 781, 1, 1, 3906, 508431, 2558556, 508431, 3906, 1, 1, 19531, 12714681, 320327931, 320327931, 12714681, 19531, 1, 1, 97656, 317886556
(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*A157832(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
|
|
|
FORMULA
|
|
|
EXAMPLE
|
1;
1, 1;
1, 6, 1;
1, 31, 31, 1;
1, 156, 806, 156, 1;
1, 781, 20306, 20306, 781, 1;
1, 3906, 508431, 2558556, 508431, 3906, 1;
1, 19531, 12714681, 320327931, 320327931, 12714681, 19531, 1,
|
|
MAPLE
|
mul( 5^i-1, i=1..n) ;
end proc:
|
|
MATHEMATICA
|
p[n_] := Product[5^i-1, {i, 1, n}]; t[n_, k_] := p[n]/(p[k]*p[n-k]); Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)
Table[QBinomial[n, k, 5], {n, 0, 10}, {k, 0, n}]//Flatten (* or *) q:= 5; 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 *)
|
|
PROG
|
(PARI) {q=5; T(n, k) = if(k==0, 1, if (k==n, 1, if (k<0 || n<k, 0, T(n-1, k-1) + q^k*T(n-1, k))))};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 27 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|