A383348 Triangle related to the partitions of n in three colors, read by rows.

9, 6, 243, 1, 243, 6561, 0, 90, 8748, 177147, 0, 15, 4860, 295245, 4782969, 0, 1, 1458, 216513, 9565938, 129140163, 0, 0, 252, 91854, 8680203, 301327047, 3486784401, 0, 0, 24, 24786, 4723920, 325241892, 9298091736, 94143178827, 0, 0, 1, 4374, 1712421, 215233605, 11622614670, 282429536481, 2541865828329

Offset: 1

Author

Michel Marcus, Apr 24 2025

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  9;
  6, 243;
  1, 243, 6561;
  0, 90, 8748, 177147;
  0, 15, 4860, 295245, 4782969;
  ...
		

References

• D. S. Gireesh and M. S. Mahadeva Naika, On 3-regular partitions in 3-colors, Indian J. Pure Appl. Math. 50 (2019), 137-148.

Links

• D. S. Gireesh and M. S. Mahadeva Naika, On 3-regular partitions in 3-colors, ResearchGate.
• B. Hemanthkumar and D. S. Gireesh, On ℓ-regular and 2-color partition triples modulo powers of 3, arXiv:2504.13507 [math.CO], 2025.

Crossrefs

Cf. A013733 (diagonal).

Programs

PARI

M(i,j) = if (j>i, return(0)); if (i==1, if (j==1, return(9))); if (i==2, if (j==1, return(6)); return(243)); if (i==3, if (j==1, return(1)); if (j==2, return(243)); return(6561)); if (i>=4, if (j==1, return(0)); 27*M(i-1,j-1) + 9*M(i-2,j-1) + M(i-3,j-1));
row(n) = vector(n, i, M(n, i));

Formula

T(i,j) = 27*T(i-1,j-1) + 9*T(i-2,j-1) + T(i-3,j-1).