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

1,1

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

Table of n, a(n) for n=1..45.

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.

Formula

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

Example

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

Prog

(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));

Crossrefs

Cf. A013733 (diagonal).

Sequence in context: A370151 A038296 A382500 * A058276 A184964 A185364

Adjacent sequences: A383345 A383346 A383347 * A383349 A383350 A383351

Keyword

nonn,tabl

Author

Michel Marcus, Apr 24 2025

Status

approved