A383348 - OEIS

%I #5 Apr 24 2025 13:21:09

%S 9,6,243,1,243,6561,0,90,8748,177147,0,15,4860,295245,4782969,0,1,

%T 1458,216513,9565938,129140163,0,0,252,91854,8680203,301327047,

%U 3486784401,0,0,24,24786,4723920,325241892,9298091736,94143178827,0,0,1,4374,1712421,215233605,11622614670,282429536481,2541865828329

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

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

%H D. S. Gireesh and M. S. Mahadeva Naika, <a href="https://www.researchgate.net/publication/325381164_On_3-Regular_Partitions_in_3-Colors">On 3-regular partitions in 3-colors</a>, ResearchGate.

%H B. Hemanthkumar and D. S. Gireesh, <a href="https://arxiv.org/abs/2504.13507">On ℓ-regular and 2-color partition triples modulo powers of 3</a>, arXiv:2504.13507 [math.CO], 2025.

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

%e Triangle begins:

%e   9;

%e   6, 243;

%e   1, 243, 6561;

%e   0, 90, 8748, 177147;

%e   0, 15, 4860, 295245, 4782969;

%e   ...

%o (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));

%o row(n) = vector(n, i, M(n, i));

%Y Cf. A013733 (diagonal).

%K nonn,tabl

%O 1,1

%A _Michel Marcus_, Apr 24 2025