A395461 Expansion of e.g.f. x^3*(1 + x + x^2/2 + x^3/6)*exp(x).

0, 0, 0, 6, 48, 240, 960, 3150, 8736, 21168, 46080, 92070, 171600, 302016, 506688, 816270, 1270080, 1917600, 2820096, 4052358, 5704560, 7884240, 10718400, 14355726, 18968928, 24757200, 31948800, 40803750, 51616656, 64719648, 80485440, 99330510, 121718400, 148163136, 179232768, 215553030

Offset

0,4

Comments

Number of words of length n defined on {a, b, c, d, e} that contain one a, one b, one c, at most three d's, and any number of e's.

Links

Table of n, a(n) for n=0..35.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = n*(n-1)*(n-2)^2*(n^2-7*n+18)/6.

a(n) = 6*binomial(n,3)*Sum_{k=0..3} binomial(n-3,k).

G.f.: 6*x^3*(1 + x + 5*x^2 + 13*x^3)/(1 - x)^7. - Stefano Spezia, May 02 2026

Mathematica

a[n_]:= n*(n-1)*(n-2)^2*(n^2-7*n+18)/6; Array[a, 36, 0] (* Stefano Spezia, May 02 2026 *)

Crossrefs

Cf. A047927, A395476.

Sequence in context: A353247 A262354 A395476 * A052771 A056289 A056284

Adjacent sequences: A395458 A395459 A395460 * A395462 A395463 A395464

Keyword

nonn,easy

Author

Enrique Navarrete, Apr 23 2026

Status

approved