Voozh

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

(list; table; graph; refs; listen; history; text; internal format)

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.

Index entries for sequences related to Gaussian binomial coefficients

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

EXAMPLE

Triangle begins:

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;

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 *)

PROG

(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

(PARI) {q=8; 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

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

Sequence in context: A166961 A202988 A098436 * A173005 A015123 A176647

Adjacent sequences: A022169 A022170 A022171 * A022173 A022174 A022175