A384686 a(n) = 2^(n-4)*(5*binomial(n,5) + 6*binomial(n,4)).

0, 0, 0, 0, 6, 70, 480, 2520, 11200, 44352, 161280, 549120, 1774080, 5491200, 16400384, 47523840, 134184960, 370442240, 1002700800, 2667184128, 6985482240, 18042716160, 46022000640, 116064256000, 289696382976, 716282265600, 1755735654400, 4269382041600, 10305404928000

Offset: 0

Author

Enrique Navarrete, Jun 07 2025

Keywords

• nonn
• easy

Comments

a(n) is the number of words of length n defined on 5 letters that have exactly two a's and exactly two b's and no c's or exactly two a's and exactly three c's and no b's.

Examples

			a(4) = 6 since the words are the 6 permutations of aabb.
a(6) = 480 since the words are the 90 permutations of aabbdd, the 180 permutations of aabbde, the 90 permutations of aabbee, the 60 permutations of aacccd, and the 60 permutations of aaccce.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Crossrefs

Cf. A384506.

Programs

Mathematica

A384686[n_] := 2^(n - 4)*(5*Binomial[n, 5] + 6*Binomial[n, 4]);
Array[A384686, 30, 0] (* Paolo Xausa, Jun 13 2025 *)
LinearRecurrence[{12,-60,160,-240,192,-64},{0,0,0,0,6,70},40] (* Harvey P. Dale, Jul 19 2025 *)

Formula

a(n) = 1/3*2^(n-7)*(n-3)*(n-2)*(n-1)*n*(n+2).

E.g.f.: x^2/2*exp(2*x)*(x^2/2 + x^3/6).

G.f.: 2*x^4*(3 - x)/(1 - 2*x)^6. - Stefano Spezia, Jun 07 2025