A369421 Expansion of o.g.f. x^3/((1-2*x)^3*(1-3*x)^2).

0, 0, 0, 1, 12, 87, 494, 2421, 10752, 44515, 174930, 660465, 2416580, 8622879, 30147798, 103653613, 351451656, 1177793499, 3908152538, 12858800745, 42001844940, 136331670295, 440083517790, 1413758295141, 4522328690192, 14411305112787, 45769389047394, 144920040864481

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 3 possible tasks;

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

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (12,-57,134,-156,72).

Formula

a(n) = 12*a(n-1) - 57*a(n-2) + 134*a(n-3) - 156*a(n-4) + 72*a(n-5), n >= 5, a(0)=a(1)=a(2)=0, a(3)=1, a(4)=12.

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

a(n) = 3^n*(n - 8) + 2^(n-2)*(n^2 + 9*n + 32). - Stefano Spezia, Jan 23 2024

Example

The following diagrams illustrate the 87 cases for n = 5, 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| x x 2 2,   24;
  x 3| x x 2,   36;
  x 3 3| x x,   27.

Mathematica

LinearRecurrence[{12, -57, 134, -156, 72}, {0, 0, 0, 1, 12}, 30] (* Paolo Xausa, Feb 23 2024 *)

Crossrefs

Cf. A369416, A369418.

Sequence in context: A091119 A243248 A046023 * A183721 A180797 A137207

Adjacent sequences: A369418 A369419 A369420 * A369422 A369423 A369424

Keyword

nonn,easy

Author

Enrique Navarrete, Jan 22 2024

Status

approved