A173749 Square array T(n, k) = v(k, n)((1)), where v(n, q) = M*v(n-1, q), M = {{0, 1, 0}, {0, 0, 1}, {q^3, q^3, 0}}, with v(0, q) = {1, 1, 1}, read by antidiagonals.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 16, 1, 1, 1, 3, 16, 54, 1, 1, 1, 4, 136, 54, 128, 1, 1, 1, 5, 256, 1485, 128, 250, 1, 1, 1, 7, 1216, 2916, 8256, 250, 432, 1, 1, 1, 9, 3136, 41553, 16384, 31375, 432, 686, 1, 1, 1

Offset

0,7

Links

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Formula

T(n, k) = v(k, n)((1)), where v(n, q) = M*v(n-1, q), M = {{0, 1, 0}, {0, 0, 1}, {q^3, q^3, 0}}, with v(0, q) = {1, 1, 1} (square array).

T(n, k) = f(k, n+1), where f(n, q) = q^3*f(n-2, q) + q^3*f(n-3, q), and f(0,q) = f(1,q) = f(2,q) = 1 (square array). - G. C. Greubel, Jul 06 2021

Example

Square array begins as:
  1, 1, 1,   2,   2,     3, ...;
  1, 1, 1,  16,  16,   136, ...;
  1, 1, 1,  54,  54,  1485, ...;
  1, 1, 1, 128, 128,  8256, ...;
  1, 1, 1, 250, 250, 31375, ...;
  1, 1, 1, 432, 432, 93528, ...;
Antidiagonal triangle begins as:
  1;
  1,    1;
  1,    1,     1;
  2,    1,     1,     1;
  2,   16,     1,     1,     1;
  3,   16,    54,     1,     1,   1;
  4,  136,    54,   128,     1,   1,   1;
  5,  256,  1485,   128,   250,   1,   1, 1;
  7, 1216,  2916,  8256,   250, 432,   1, 1, 1;
  9, 3136, 41553, 16384, 31375, 432, 686, 1, 1, 1;

Mathematica

(* First program *)
M = {{0, 1, 0}, {0, 0, 1}, {q^3, q^3, 0}};
v[0, q_] = {1, 1, 1};
v[n_, q_]:= v[n, q]= M.v[n-1, q];
T = Table[v[n, q][[1]], {n, 0, 20}, {q, 1, 21}];
Table[T[[n-k+1, k+1]], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jul 06 2021 *)
(* Second program *)
f[n_, q_]:= f[n, q] = If[n<3, 1, q^3*f[n-2, q] + q^3*f[n-3, q]];
T[n_, k_]:= f[k, n+1];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 06 2021 *)

Prog

(Magma)
function t(n, k)
  if n lt 3 then return 1;
  else return k^3*t(n-2, k) + k^3*t(n-3, k);
  end if; return t;
end function;
[t(n-k, k+1): 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): return f(k, n+1)
flatten([[T(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 06 2021

Crossrefs

Cf. A173747, A173778, A173779.

Sequence in context: A231867 A105685 A228239 * A323618 A364650 A246270

Adjacent sequences: A173746 A173747 A173748 * A173750 A173751 A173752

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 23 2010

Extensions

Edited by G. C. Greubel, Jul 06 2021

Status

approved