A022169 Triangle of Gaussian binomial coefficients [ n,k ] for q = 5.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl

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

Examples

			Triangle begins:
  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,
		

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

• 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

Crossrefs

Cf. A003462 (column k=1), A006111 (k=2), A006112 (k=3).

Row sums give A006119.

Programs

Maple

A027872 := proc(n)
        mul( 5^i-1, i=1..n) ;
end proc:
A022169 := proc(n, m)
        A027872(n)/A027872(n-m)/A027872(m) ;
end proc: # R. J. Mathar, Mar 12 2013

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 *)

PARI

{q=5; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

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) = 5^j - 1. - Seiichi Manyama, May 09 2025