A369416 Expansion of o.g.f. x^4/((1-2*x)^3*(1-3*x)^3).

0, 0, 0, 0, 1, 15, 132, 890, 5091, 26025, 122590, 542700, 2288565, 9282275, 36469704, 139556910, 522324343, 1918424685, 6933067554, 24707355200, 86980866345, 302944443975, 1045165002220, 3575578524450, 12140493868491, 40943810295665, 137242735999782

Offset

0,6

Comments

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

1) in the first interval choose 2 subintervals 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

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

Index entries for linear recurrences with constant coefficients, signature (15,-93,305,-558,540,-216).

Formula

a(n) = 15*a(n-1) - 93*a(n-2) + 305*a(n-3) - 558*a(n-4) + 540*a(n-5) - 216*a(n-6), n >= 6, a(0)=a(1)=a(2)=a(3)=0, a(4)=1, a(5)=15.

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

a(n) = 3^n * (n^2/6 - 17*n/6 + 15) - 2^n * (n^2/4 + 13*n/4 + 15). - Robert Israel, Jan 22 2024

Example

The following diagrams illustrate the 890 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 x 2 2 2 | x x,    80;
    x x 2 2 | x x 3,   216;
    x x 2 | x x 3 3,   324;
    x x| x x 3 3 3,    270.

Maple

seq( 3^n * (n^2/6 - 17 * n/6 + 15) - 2^n * (n^2/4 + 13*n/4 + 15), n=0..40); # Robert Israel, Jan 22 2024

Mathematica

LinearRecurrence[{15, -93, 305, -558, 540, -216}, {0, 0, 0, 0, 1, 15}, 30] (* Paolo Xausa, Feb 23 2024 *)

Crossrefs

Cf. A369418, A369421.

Sequence in context: A275101 A236262 A227814 * A236259 A348212 A289421

Adjacent sequences: A369413 A369414 A369415 * A369417 A369418 A369419

Keyword

nonn,easy

Author

Enrique Navarrete, Jan 22 2024

Status

approved