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A173119 Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 3, read by rows. 6
1, 1, 1, 1, 5, 1, 1, 6, 6, 1, 1, 7, 12, 7, 1, 1, 8, 28, 28, 8, 1, 1, 9, 36, 56, 36, 9, 1, 1, 10, 45, 119, 119, 45, 10, 1, 1, 11, 55, 164, 238, 164, 55, 11, 1, 1, 12, 66, 219, 483, 483, 219, 66, 12, 1, 1, 13, 78, 285, 702, 966, 702, 285, 78, 13, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 3.
Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = 3. - G. C. Greubel, Apr 27 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 5, 1;
1, 6, 6, 1;
1, 7, 12, 7, 1;
1, 8, 28, 28, 8, 1;
1, 9, 36, 56, 36, 9, 1;
1, 10, 45, 119, 119, 45, 10, 1;
1, 11, 55, 164, 238, 164, 55, 11, 1;
1, 12, 66, 219, 483, 483, 219, 66, 12, 1;
1, 13, 78, 285, 702, 966, 702, 285, 78, 13, 1;
MATHEMATICA
T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j, 0, 5}]];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)
PROG
(Sage)
@CachedFunction
def T(n, k, q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Apr 27 2021
CROSSREFS
Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), A173118 (q=2), this sequence (q=3), A173120 (q= -4), A173122.
Sequence in context: A280374 A259975 A028313 * A050178 A297986 A298635
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Feb 10 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 27 2021
STATUS
approved

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Last modified April 18 15:48 EDT 2024. Contains 371780 sequences. (Running on oeis4.)