A173048 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 = 2, read by rows.

1, 3, 3, 9, 12, 9, 65, 70, 70, 65, 1025, 990, 560, 990, 1025, 32769, 31806, 11160, 11160, 31806, 32769, 2097153, 2064510, 671832, 178560, 671832, 2064510, 2097153, 268435457, 266338558, 87413592, 12850368, 12850368, 87413592, 266338558, 268435457

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 = 2.

Example

Triangle begins as:
        1;
        3,       3;
        9,      12,      9;
       65,      70,     70,     65;
     1025,     990,    560,    990,   1025;
    32769,   31806,  11160,  11160,  31806,   32769;
  2097153, 2064510, 671832, 178560, 671832, 2064510, 2097153;

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, 2], {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, 2): n in [0..10]]; // G. C. Greubel, Apr 26 2021

Crossrefs

Cf. A134058 (q=1), this sequence (q=2), A173049 (q=3).

Sequence in context: A320293 A146153 A339028 * A165992 A117153 A045810

Adjacent sequences: A173045 A173046 A173047 * A173049 A173050 A173051

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Feb 08 2010

Extensions

Edited by G. C. Greubel, Apr 26 2021

Status

approved