A383136 a(n) = Sum_{k=0..n} k^2 * 2^(n-k) * binomial(n,k).

0, 1, 8, 45, 216, 945, 3888, 15309, 58320, 216513, 787320, 2814669, 9920232, 34543665, 119042784, 406552365, 1377495072, 4634696961, 15496819560, 51526925037, 170465015160, 561372288561, 1841022163728, 6014703091725, 19581781196016, 63546645708225, 205608702558168

Offset

0,3

Comments

a(n) is the number of length n words on the alphabet {a,b,c,d,e} such that each word contains at most one a and exactly one b. - Enrique Navarrete, Nov 28 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Formula

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

G.f.: x*(1 - x)/(1 - 3*x)^3. - Andrew Howroyd, Nov 12 2025

From Enrique Navarrete, Nov 27 2025: (Start)

a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3).

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

From Amiram Eldar, Dec 04 2025: (Start)

Sum_{n>=1} 1/a(n) = 63/4 - 36*log(3/2).

Sum_{n>=1} (-1)^(n+1)/a(n) = 45/4 - 36*log(4/3). (End)

Example

a(2) = 8 since the words are (number of permutations in parentheses): ab (2), bc (2), bd (2), be (2). - Enrique Navarrete, Nov 28 2025

Mathematica

a[n_] := 3^(n-2) * n * (2 + n); Array[a, 27, 0] (* Amiram Eldar, Dec 04 2025 *)

Prog

(PARI) a(n) = 3^(n-2)*n*(2+n);

Crossrefs

Cf. A027471, A383137, A383138, A383139.

First differences of A027472; second differences of A367591.

Sequence in context: A273267 A055222 A273305 * A026015 A002696 A016208

Adjacent sequences: A383133 A383134 A383135 * A383137 A383138 A383139

Keyword

nonn,easy

Author

Seiichi Manyama, Apr 17 2025

Status

approved