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A022174 Triangle of Gaussian binomial coefficients [ n,k ] for q = 10. 17
1, 1, 1, 1, 11, 1, 1, 111, 111, 1, 1, 1111, 11211, 1111, 1, 1, 11111, 1122211, 1122211, 11111, 1, 1, 111111, 112232211, 1123333211, 112232211, 111111, 1, 1, 1111111, 11223332211, 1123445443211, 1123445443211 (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

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 11 1;

1 111 111 1;

1 1111 11211 1111 1;

1 11111 1122211 1122211 11111 1;

1 111111 112232211 1123333211 112232211 111111 1;

1 1111111 11223332211 1123445443211 1123445443211 11223332211 1111111 1;

MAPLE

A027878 := proc(n)

    mul(10^i-1, i=1..n) ;

end proc:

A022174 := proc(n, m)

    A027878(n)/A027878(m)/A027878(n-m) ;

end proc:# R. J. Mathar, Jul 19 2017

MATHEMATICA

a027878[n_]:=Times@@ Table[10^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, 10], {n, 0, 10}, {k, 0, n}]//Flatten (* or *) q:= 10; 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 28 2018 *)

PROG

(Python)

from operator import mul

def a027878(n): return 1 if n==0 else reduce(mul, [10**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=10; 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 28 2018

CROSSREFS

Row sums give A015196.

Sequence in context: A014469 A157149 A166979 * A173006 A015125 A290552

Adjacent sequences:  A022171 A022172 A022173 * A022175 A022176 A022177

KEYWORD

nonn,tabl,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 13 06:26 EST 2019. Contains 329968 sequences. (Running on oeis4.)