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.

1, 4, 4, 28, 24, 28, 730, 390, 390, 730, 59050, 29280, 7020, 29280, 59050, 14348908, 7145292, 914760, 914760, 7145292, 14348908, 10460353204, 5223003240, 650485836, 49397040, 650485836, 5223003240, 10460353204, 22876792454962, 11433166054158, 1427188022442, 55340738838, 55340738838, 1427188022442, 11433166054158, 22876792454962

Offset: 0

Author

Roger L. Bagula, Feb 08 2010

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle begins as:
            1;
            4,          4;
           28,         24,        28;
          730,        390,       390,      730;
        59050,      29280,      7020,    29280,     59050;
     14348908,    7145292,    914760,   914760,   7145292,   14348908;
  10460353204, 5223003240, 650485836, 49397040, 650485836, 5223003240, 10460353204;
		

Links

• G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Crossrefs

Cf. A134058 (q=1), A173048 (q=2), this sequence (q=3).

Programs

Magma

R:=PowerSeriesRing(Integers(), 50);
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]]) >;
T:= func< n,q | Coefficients(R!( p(x,n,q) )) >;
[T(n,3): n in [0..10]]; // G. C. Greubel, Apr 26 2021

Mathematica

p[x_, n_, q_]:= If[n==0, 1, Product[x+q^j, {j,n}] + Product[x*q^j +1, {j,n}]];
T[n_, k_, q_]:= SeriesCoefficient[p[x,n,q], {x,0,k}];
Table[T[n, k, 3], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 26 2021 *)

Formula

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.

Extensions

Edited by G. C. Greubel, Apr 26 2021