A022173 Triangle of Gaussian binomial coefficients [ n,k ] for q = 9.

1, 1, 1, 1, 10, 1, 1, 91, 91, 1, 1, 820, 7462, 820, 1, 1, 7381, 605242, 605242, 7381, 1, 1, 66430, 49031983, 441826660, 49031983, 66430, 1, 1, 597871, 3971657053, 322140667123, 322140667123, 3971657053

Offset

0,5

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.

Formula

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017

Example

1 ;
1 1;
1 10 1;
1 91 91 1;
1 820 7462 820 1;
1 7381 605242 605242 7381 1;
1 66430 49031983 441826660 49031983 66430 1;
1 597871 3971657053 322140667123 322140667123 3971657053 597871 1;
1 5380840 321704819164 234844517989720 2113887057661126 234844517989720 321704819164 5380840 1 ;

Maple

A027877 := proc(n)
    mul(9^i-1, i=1..n) ;
end proc:
A022173 := proc(n, m)
    A027877(n)/A027877(m)/A027877(n-m) ;
end proc: # R. J. Mathar, Jul 19 2017

Mathematica

a027878[n_]:=Times@@ Table[9^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, after Maple code *)
Table[QBinomial[n, k, 9], {n, 0, 10}, {k, 0, n}]//Flatten (* or *) q:= 9; 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

(Python)
from operator import mul
def a027878(n): return 1 if n==0 else reduce(mul, [9**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, after Maple code
(PARI) {q=9; 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

Sequence in context: A364071 A160562 A176243 * A158117 A172378 A015124

Adjacent sequences: A022170 A022171 A022172 * A022174 A022175 A022176

Keyword

nonn,tabl

Author

N. J. A. Sloane

Status

approved