A173778 Triangle T(n, k, q) = round(c(n,q)/(c(k,q)*c(n-k,q))), where c(n,q) = Product_{j=0..n} v(j, q)((1)), v(n, q) = M*v(n-1, q), v(0, q) = {1, 1, 1}, M = {{0, 1, 0}, {0, 0, 1}, {q^3, q^3, 0}}, and q = 2, read by rows.

1, 1, 1, 1, 1, 1, 1, 16, 16, 1, 1, 16, 256, 16, 1, 1, 136, 2176, 2176, 136, 1, 1, 256, 34816, 34816, 34816, 256, 1, 1, 1216, 311296, 2646016, 2646016, 311296, 1216, 1, 1, 3136, 3813376, 61014016, 518619136, 61014016, 3813376, 3136, 1, 1, 11776, 36929536, 2806644736, 44906315776, 44906315776, 2806644736, 36929536, 11776, 1

Offset

0,8

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k, q) = round( c(n,q)/(c(k,q)*c(n-k,q)) ), where c(n,q) = Product_{j=0..n} v(j, q)((1)), v(n, q) = M*v(n-1, q), v(0, q) = {1, 1, 1}, M = {{0, 1, 0}, {0, 0, 1}, {q^3, q^3, 0}}, and q = 2.

T(n, k, q) = round( (1/f(k,q))*Product_{j=0..n-k} f(j+k,q)/f(j,q) ), where f(n, q) = 6*q*f(n-2, q) + 8*q^3*f(n-3, q), and f(0,q) = f(1,q) = f(2,q) = 1, and q = 2. - G. C. Greubel, Jul 06 2021

Example

Triangle begins as:
  1;
  1,    1;
  1,    1,       1;
  1,   16,      16,        1;
  1,   16,     256,       16,         1;
  1,  136,    2176,     2176,       136,        1;
  1,  256,   34816,    34816,     34816,      256,       1;
  1, 1216,  311296,  2646016,   2646016,   311296,    1216,    1;
  1, 3136, 3813376, 61014016, 518619136, 61014016, 3813376, 3136, 1;

Mathematica

f[n_, q_]:= f[n, q] = If[n<3, 1, q^3*f[n-2, q] + q^3*f[n-3, q]];
c[n_, q_]:= Product[f[j, q], {j, 0, n}];
T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 06 2021 *)

Prog

(Magma)
function f(n, k)
  if n lt 3 then return 1;
  else return k^3*f(n-2, k) + k^3*f(n-3, k);
  end if; return f;
end function;
T:= func< n, k, q | Round( (&*[f(j+k, q)/f(j, q): j in [0..n-k]])/f(k, q) ) >;
[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 06 2021
(Sage)
@CachedFunction
def f(n, q): return 1 if (n<3) else q^3*f(n-2, q) + q^3*f(n-3, q)
def T(n, k, q): return round( product( f(j+k, q)/f(j, q) for j in (0..n-k))/f(k, q) )
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 06 2021

Crossrefs

Cf. this sequence (q=2), A173779 (q=4).

Cf. A173747, A173749, A173779.

Sequence in context: A023458 A284498 A070576 * A103908 A192632 A305945

Adjacent sequences: A173775 A173776 A173777 * A173779 A173780 A173781

Keyword

nonn,tabl,less

Author

Roger L. Bagula, Feb 24 2010

Extensions

Definition corrected and edited by G. C. Greubel, Jul 06 2021

Status

approved