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}, {8*q^3, 6*q, 0}}, with v(0, q) = {1, 1, 1} (square array).
T(n, k) = f(k, n+1), 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 (square array). - G. C. Greubel, Jul 06 2021
EXAMPLE
Square array begins as:
1, 1, 1, 14, 14, 92, ...;
1, 1, 1, 76, 76, 976, ...;
1, 1, 1, 234, 234, 4428, ...;
1, 1, 1, 536, 536, 13376, ...;
1, 1, 1, 1030, 1030, 31900, ...;
1, 1, 1, 1764, 1764, 65232, ...;
Antidiagonal triangle begins as:
1;
1, 1;
1, 1, 1;
14, 1, 1, 1;
14, 76, 1, 1, 1;
92, 76, 234, 1, 1, 1;
196, 976, 234, 536, 1, 1, 1;
664, 5776, 4428, 536, 1030, 1, 1, 1;
1912, 16576, 54756, 13376, 1030, 1764, 1, 1, 1;
5552, 131776, 130248, 287296, 31900, 1764, 2786, 1, 1, 1;
MATHEMATICA
(* First program *)
M = {{0, 1, 0}, {0, 0, 1}, {8*q^3, 6*q, 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, 6*q*f[n-2, q] + 8*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
(Sage)
@CachedFunction
def f(n, q): return 1 if (n<3) else 6*q*f(n-2, q) + 8*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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 23 2010
EXTENSIONS
Edited by G. C. Greubel, Jul 06 2021
STATUS
approved