A022172 Triangle of Gaussian binomial coefficients [ n,k ] for q = 8.

1, 1, 1, 1, 9, 1, 1, 73, 73, 1, 1, 585, 4745, 585, 1, 1, 4681, 304265, 304265, 4681, 1, 1, 37449, 19477641, 156087945, 19477641, 37449, 1, 1, 299593, 1246606473, 79936505481, 79936505481, 1246606473, 299593, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,      1;
  1,      9,          1;
  1,     73,         73,           1;
  1,    585,       4745,         585,           1;
  1,   4681,     304265,      304265,        4681,          1;
  1,  37449,   19477641,   156087945,    19477641,      37449,      1;
  1, 299593, 1246606473, 79936505481, 79936505481, 1246606473, 299593, 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

Crossrefs

Cf. A023001 (k=1), A022242 (k=2).

Programs

Maple

A027876 := proc(n)
    mul(8^i-1,i=1..n) ;
end proc:
A022172 := proc(n,m)
    A027876(n)/A027876(m)/A027876(n-m) ;
end proc: # R. J. Mathar, Jul 19 2017

Mathematica

a027878[n_]:=Times@@ Table[8^i - 1, {i, n}]; T[n_, m_]:=a027878[n]/( a027878[m] a027878[n - m]); Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* Indranil Ghosh, Jul 20 2017 *)
Table[QBinomial[n,k,8], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 8; 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=8; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

Python

from operator import mul
def a027878(n): return 1 if n==0 else reduce(mul, [8**i - 1 for i in range(1, n + 1)])
def T(n, m): return a027878(n)//(a027878(m)*a027878(n - m))
for n in range(11): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, Jul 20 2017

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) = 8^j - 1. - Seiichi Manyama, May 09 2025