A383910 Expansion of Product_{k=0..3} (1 + k*x)/(1 - k*x).

1, 12, 72, 312, 1152, 3912, 12672, 39912, 123552, 378312, 1150272, 3481512, 10505952, 31640712, 95167872, 285995112, 858968352, 2578871112, 7740545472, 23229500712, 69704230752, 209144149512, 627495363072, 1882611918312, 5648087413152, 16944765555912, 50835303300672

Offset: 0

Author

Seiichi Manyama, May 14 2025

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Column k=3 of A383900.

Cf. A091344, A383912.

Programs

PARI

a(n) = if(n==0, 1, 20*3^n-15*2^(n+1)+12);

Formula

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.

a(n) = Sum_{k=0..3} |Stirling1(4,k+1)| * Stirling2(k+n,3).

a(n) = A383912(n) + 3*A383912(n-1) = 20*3^n - 15*2^(n+1) + 12 = 10*A091344(n) + 2 for n > 0.