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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

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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

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

EXAMPLE

Triangle begins:

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

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 *)

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

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

Row sums give A006119.

Sequence in context: A166349 A176429 A157155 * A156601 A178232 A203338

Adjacent sequences: A022166 A022167 A022168 * A022170 A022171 A022172

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane

STATUS

approved

URL: https://oeis.org/A022169

⇱ A022169 - OEIS