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%I #12 Sep 08 2022 08:45:50
%S 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,
%T 256,1485,128,250,1,1,1,7,1216,2916,8256,250,432,1,1,1,9,3136,41553,
%U 16384,31375,432,686,1,1,1
%N 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.
%H G. C. Greubel, <a href="/A173749/b173749.txt">Antidiagonal rows n = 0..50, flattened</a>
%F 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).
%F 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
%e Square array begins as:
%e 1, 1, 1, 2, 2, 3, ...;
%e 1, 1, 1, 16, 16, 136, ...;
%e 1, 1, 1, 54, 54, 1485, ...;
%e 1, 1, 1, 128, 128, 8256, ...;
%e 1, 1, 1, 250, 250, 31375, ...;
%e 1, 1, 1, 432, 432, 93528, ...;
%e Antidiagonal triangle begins as:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 2, 1, 1, 1;
%e 2, 16, 1, 1, 1;
%e 3, 16, 54, 1, 1, 1;
%e 4, 136, 54, 128, 1, 1, 1;
%e 5, 256, 1485, 128, 250, 1, 1, 1;
%e 7, 1216, 2916, 8256, 250, 432, 1, 1, 1;
%e 9, 3136, 41553, 16384, 31375, 432, 686, 1, 1, 1;
%t (* First program *)
%t M = {{0, 1, 0}, {0, 0, 1}, {q^3, q^3, 0}};
%t v[0, q_] = {1, 1, 1};
%t v[n_, q_]:= v[n, q]= M.v[n-1, q];
%t T = Table[v[n, q][[1]], {n,0,20}, {q,1,21}];
%t Table[T[[n-k+1, k+1]], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jul 06 2021 *)
%t (* Second program *)
%t f[n_, q_]:= f[n, q] = If[n<3, 1, q^3*f[n-2, q] + q^3*f[n-3, q]];
%t T[n_, k_]:= f[k, n+1];
%t Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 06 2021 *)
%o (Magma)
%o function t(n,k)
%o if n lt 3 then return 1;
%o else return k^3*t(n-2,k) + k^3*t(n-3,k);
%o end if; return t;
%o end function;
%o [t(n-k,k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 06 2021
%o (Sage)
%o @CachedFunction
%o def f(n,q): return 1 if (n<3) else q^3*f(n-2, q) + q^3*f(n-3, q)
%o def T(n,k): return f(k, n+1)
%o flatten([[T(k, n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 06 2021
%Y Cf. A173747, A173778, A173779.
%K nonn,tabl
%O 0,7
%A _Roger L. Bagula_, Feb 23 2010
%E Edited by _G. C. Greubel_, Jul 06 2021
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