A384852 a(n) = 2*binomial(n,2) + 3*binomial(n,3) + 4*binomial(n,4).

0, 0, 2, 9, 28, 70, 150, 287, 504, 828, 1290, 1925, 2772, 3874, 5278, 7035, 9200, 11832, 14994, 18753, 23180, 28350, 34342, 41239, 49128, 58100, 68250, 79677, 92484, 106778, 122670, 140275, 159712, 181104, 204578, 230265, 258300, 288822, 321974, 357903, 396760

Offset

0,3

Comments

a(n) is the number of words of length n defined on 5 letters that contain one a (and no b's or c's), two b's (and no a's or c's) or three c's (and no a's or b's), and, in all cases, one d and any number of e's.

Links

Jakub Buczak, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 1/6*n^4 - 1/2*n^3 + 4/3*n^2 - n.

G.f.: x^2*(3*x^2 - x + 2)/(1 - x)^5.

E.g.f.: exp(x)*(x^2 + x^3/2 + x^4/6).

Example

a(4) = 28 since the words are the 12 permutations of adee, the 12 permutations of bbde, and the 4 permutations of cccd.
a(5) = 70 since the words are the 20 permutations of adeee, the 30 permutations of bbdee, and the 20 permutations of cccde.

Mathematica

a[n_] := Sum[k * Binomial[n, k], {k, 2, 4}]; Array[a, 50, 0] (* Amiram Eldar, Jun 22 2025 *)

Crossrefs

Cf. A004006.

Sequence in context: A121643 A353017 A183376 * A131066 A341507 A058877

Adjacent sequences: A384849 A384850 A384851 * A384853 A384854 A384855

Keyword

nonn,easy

Author

Enrique Navarrete, Jun 19 2025

Status

approved