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

0,2

Links

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

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.

Example

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;

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 *)

Prog

(Magma)
R<x>:=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

Crossrefs

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

Sequence in context: A078146 A066836 A227715 * A272040 A338672 A065237

Adjacent sequences: A173046 A173047 A173048 * A173050 A173051 A173052

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Feb 08 2010

Extensions

Edited by G. C. Greubel, Apr 26 2021

Status

approved