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A173049 Triangle T(n, k) = [x^k](p(x, n, q)) where p(x,n,q) = Product_{j=1..n} (x + q^j) + Product_{j=1..n} (x*q^j + 1), p(x, 0, q) = 1, and q = 3, read by rows.

%I A173049 #7 Apr 27 2021 01:05:41
%S A173049 1,4,4,28,24,28,730,390,390,730,59050,29280,7020,29280,59050,14348908,
%T A173049 7145292,914760,914760,7145292,14348908,10460353204,5223003240,
%U A173049 650485836,49397040,650485836,5223003240,10460353204,22876792454962,11433166054158,1427188022442,55340738838,55340738838,1427188022442,11433166054158,22876792454962
%N A173049 Triangle T(n, k) = [x^k](p(x, n, q)) where p(x,n,q) = Product_{j=1..n} (x + q^j) + Product_{j=1..n} (x*q^j + 1), p(x, 0, q) = 1, and q = 3, read by rows.
%H A173049 G. C. Greubel, <a href="/A173049/b173049.txt">Rows n = 0..25 of the triangle, flattened</a>
%F A173049 T(n, k) = [x^k](p(x, n, q)) where p(x,n,q) = Product_{j=1..n} (x + q^j) + Product_{j=1..n} (x*q^j + 1), p(x, 0, q) = 1, and q = 3.
%e A173049 Triangle begins as:
%e A173049             1;
%e A173049             4,          4;
%e A173049            28,         24,        28;
%e A173049           730,        390,       390,      730;
%e A173049         59050,      29280,      7020,    29280,     59050;
%e A173049      14348908,    7145292,    914760,   914760,   7145292,   14348908;
%e A173049   10460353204, 5223003240, 650485836, 49397040, 650485836, 5223003240, 10460353204;
%t A173049 p[x_, n_, q_]:= If[n==0, 1, Product[x+q^j, {j,n}] + Product[x*q^j +1, {j,n}]];
%t A173049 T[n_, k_, q_]:= SeriesCoefficient[p[x,n,q], {x,0,k}];
%t A173049 Table[T[n, k, 3], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 26 2021 *)
%o A173049 (Magma)
%o A173049 R<x>:=PowerSeriesRing(Integers(), 50);
%o A173049 p:= func< x,n,q | n eq 0 select 1 else (&*[x+q^j: j in [1..n]]) + (&*[1+q^j*x: j in [1..n]]) >;
%o A173049 T:= func< n,q | Coefficients(R!( p(x,n,q) )) >;
%o A173049 [T(n,3): n in [0..10]]; // _G. C. Greubel_, Apr 26 2021
%Y A173049 Cf. A134058 (q=1), A173048 (q=2), this sequence (q=3).
%K A173049 nonn,tabl,easy,less
%O A173049 0,2
%A A173049 _Roger L. Bagula_, Feb 08 2010
%E A173049 Edited by _G. C. Greubel_, Apr 26 2021