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

0,5

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.

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).

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

From Amiram Eldar, Sep 29 2025: (Start)

Sum_{n>=4} 1/a(n) = 12*log(2)/5 - 37/25.

Sum_{n>=4} (-1)^n/a(n) = 324*log(3/2)/5 - 653/25. (End)

Example

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.

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 *)

Crossrefs

Cf. A384506.

Sequence in context: A188406 A048708 A286527 * A104900 A186667 A001448

Adjacent sequences: A384683 A384684 A384685 * A384687 A384688 A384689

Keyword

nonn,easy

Author

Enrique Navarrete, Jun 07 2025

Status

approved