A369418 Expansion of o.g.f. x^3/((1-2*x)^2*(1-3*x)^3).

0, 0, 0, 1, 13, 102, 626, 3311, 15843, 70540, 297520, 1203165, 4705145, 17905154, 66617502, 243210523, 873775999, 3096218184, 10841220092, 37566155945, 128982711285, 439276114270, 1485248520010, 4989336819591, 16662822558683, 55355115408452, 183012125047176, 602417637911221

Offset

0,5

Comments

a(n) is the number of ways to split [n] into 2 intervals and perform the following:

1) in the first interval choose one subinterval and in the remaining subintervals (if there are any left), perform one of 2 possible tasks;

2) in the second interval choose 2 subintervals and in the remaining subintervals (if there are any left), perform one of 3 possible tasks.

Links

Table of n, a(n) for n=0..27.

Index entries for linear recurrences with constant coefficients, signature (13,-67,171,-216,108).

Formula

a(n) = 13*a(n-1) - 67*a(n-2) + 171*a(n-3) - 216*a(n-4) + 108*a(n-5), n >= 5, a(0)=a(1)=a(2)=0, a(3)=1, a(4)=13.

a(n) = Sum_{k=2..n-2} k*2^(k-1)*(n-k,2)*3^(n-k-2).

a(n) = 3^(n-1)*(n^2 - 11*n + 42)/2 - 2^n*(n + 7). - Stefano Spezia, Jan 23 2024

Example

The following diagrams illustrate the 3311 cases for n = 7, where the bar separates intervals, x represents a chosen subinterval, and i represents i possible tasks that can be performed in the subinterval, i = 2,3.
 Case, number of cases:
  x 2 2 2 2 | x x,   80;
  x 2 2 2 | x x 3,  288;
  x 2 2 | x x 3 3,  648;
  x 2 | x x 3 3 3, 1080;
  x | x x 3 3 3 3, 1215.

Mathematica

LinearRecurrence[{13, -67, 171, -216, 108}, {0, 0, 0, 1, 13}, 28] (* James C. McMahon, Feb 12 2024 *)

Crossrefs

Cf. A369416, A369421.

Sequence in context: A087595 A117653 A004635 * A289420 A050670 A142318

Adjacent sequences: A369415 A369416 A369417 * A369419 A369420 A369421

Keyword

nonn,easy

Author

Enrique Navarrete, Jan 22 2024

Status

approved