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%I #11 Sep 08 2022 08:45:50
%S 1,1,1,1,1,1,1,128,128,1,1,128,16384,128,1,1,8256,1056768,1056768,
%T 8256,1,1,16384,135266304,135266304,135266304,16384,1,1,536576,
%U 8791261184,567036346368,567036346368,8791261184,536576,1,1,1576960,846158888960,108308337786880,6985887787253760,108308337786880,846158888960,1576960,1
%N 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 = 4, read by rows.
%H G. C. Greubel, <a href="/A173779/b173779.txt">Rows n = 0..50 of the triangle, flattened</a>
%F 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 = 4.
%F 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 = 4. - _G. C. Greubel_, Jul 06 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 128, 128, 1;
%e 1, 128, 16384, 128, 1;
%e 1, 8256, 1056768, 1056768, 8256, 1;
%e 1, 16384, 135266304, 135266304, 135266304, 16384, 1;
%e 1, 536576, 8791261184, 567036346368, 567036346368, 8791261184, 536576, 1;
%t f[n_, q_]:= f[n, q] = If[n<3, 1, q^3*f[n-2, q] + q^3*f[n-3, q]];
%t c[n_, q_]:= Product[f[j, q], {j, 0, n}];
%t T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
%t Table[T[n, k, 4], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 06 2021 *)
%o (Magma)
%o function f(n,k)
%o if n lt 3 then return 1;
%o else return k^3*f(n-2,k) + k^3*f(n-3,k);
%o end if; return f;
%o end function;
%o T:= func< n,k,q | Round( (&*[f(j+k, q)/f(j,q): j in [0..n-k]])/f(k,q) ) >;
%o [T(n,k,4): 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,q): return round(product( f(j+k,q)/f(j,q) for j in (0..n-k))/f(k,q))
%o flatten([[T(n,k,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 06 2021
%Y Cf. A173778 (q=2), this sequence (q=4).
%Y Cf. A173747, A173749, A173779.
%K nonn,tabl,less
%O 0,8
%A _Roger L. Bagula_, Feb 24 2010
%E Definition corrected and edited by _G. C. Greubel_, Jul 06 2021
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