A139459 Triangle read by rows: binomial(3*n,3*k), 0 <= k <= n.

1, 1, 1, 1, 20, 1, 1, 84, 84, 1, 1, 220, 924, 220, 1, 1, 455, 5005, 5005, 455, 1, 1, 816, 18564, 48620, 18564, 816, 1, 1, 1330, 54264, 293930, 293930, 54264, 1330, 1, 1, 2024, 134596, 1307504, 2704156, 1307504, 134596, 2024, 1, 1, 2925, 296010, 4686825, 17383860, 17383860, 4686825, 296010, 2925, 1

Offset: 0

Author

Gary W. Adamson, Apr 22 2008

Keywords

• nonn
• easy
• tabl

Comments

ConvOffsStoT transform of the dodecahedral numbers A006566 starting (1, 20, 84, 220,...).

Row sums give A007613.

The matrix inverse starts:

1;

-1,1;

19,-20,1;

-1513,1596,-84,1;

315523,-332860,17556,-220,1;

-136085041,143562965,-7572565,95095,-455,1;

105261234643,-111045393456,5857368972,-73562060,352716,-816,1; - R. J. Mathar, Mar 22 2013

Examples

			First few rows of the triangle are:
  [0] 1;
  [1] 1,   1;
  [2] 1,  20,     1;
  [3] 1,  84,    84,     1;
  [4] 1, 220,   924,   220,     1;
  [5] 1, 455,  5005,  5005,   455,   1;
  [6] 1, 816, 18564, 48620, 18564, 816, 1;
  ...
Row 5 = (1, 220, 924, 220, 1) = ConvOffs transform of (1, 20, 84, 220); where A006566 = (0, 1, 20, 84, 220, 455, ...).
		

Crossrefs

Cf. A006566, A007613, A086645, A034839, A070775, A177808.

Programs

Mathematica

Table[Binomial[3*n, 3*k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 01 2025 *)

Extensions

More terms from Amiram Eldar, Jun 01 2025