A369416 - OEIS

%I #17 Feb 23 2024 06:55:53

%S 0,0,0,0,1,15,132,890,5091,26025,122590,542700,2288565,9282275,

%T 36469704,139556910,522324343,1918424685,6933067554,24707355200,

%U 86980866345,302944443975,1045165002220,3575578524450,12140493868491,40943810295665,137242735999782

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

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

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

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

%H Paolo Xausa, <a href="/A369416/b369416.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (15,-93,305,-558,540,-216).

%F 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.

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

%F 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

%e 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.

%e   Case, number of cases:

%e     x x 2 2 2 | x x,    80;

%e     x x 2 2 | x x 3,   216;

%e     x x 2 | x x 3 3,   324;

%e     x x| x x 3 3 3,    270.

%p 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

%t LinearRecurrence[{15, -93, 305, -558, 540, -216}, {0, 0, 0, 0, 1, 15}, 30] (* _Paolo Xausa_, Feb 23 2024 *)

%Y Cf. A369418, A369421.

%K nonn,easy

%O 0,6

%A _Enrique Navarrete_, Jan 22 2024