A382081 a(n) = binomial(n,3) + 6*binomial(n,4) + 15*binomial(n,5) + 15*binomial(n,6).

0, 0, 0, 1, 10, 55, 215, 665, 1736, 3990, 8310, 16005, 28930, 49621, 81445, 128765, 197120, 293420, 426156, 605625, 844170, 1156435, 1559635, 2073841, 2722280, 3531650, 4532450, 5759325, 7251426, 9052785, 11212705, 13786165, 16834240, 20424536, 24631640, 29537585

Offset

0,5

Comments

6*a(n) is the number of strings of length n defined on {0, 1, 2, 3} that contain one or two 0's, one or two 1's, one or two 2's and there are no restrictions on the number of 3's.

For n >= 6, also the number of induced cubes in the n-tetrahedral (Johnson) graph. - Eric W. Weisstein, Mar 14 2026

Links

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

Eric Weisstein's World of Mathematics, Cube Polynomial.

Eric Weisstein's World of Mathematics, Tetrahedral Johnson Graph.

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

Formula

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

G.f.: x^3*(1 + 3*x+ 6*x^2 + 5*x^3)/(1 - x)^7. - Stefano Spezia, Mar 15 2025

a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Eric W. Weisstein, Mar 14 2026

Example

6*a(4) = 60 since the strings are the 12 permutations of 0012, the 12 permutations of 0112, the 12 permutations of 0122, and the 24 permutations of 0123.

Mathematica

a[n_] := Total[{1, 6, 15, 15} Binomial[n, {3, 4, 5, 6}]]; Array[a, 36, 0] (* Amiram Eldar, Mar 15 2025 *)
Table[{1, 6, 15, 15} . Binomial[n, Range[3, 6]], {n, 0, 20}] (* Eric W. Weisstein, Mar 14 2026 *)
Table[(n - 2) (n - 1) n (n^3 - 6 n^2 + 17 n - 16)/48, {n, 0, 10}] (* Eric W. Weisstein, Mar 14 2026 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 1, 10, 55, 215, 665}, {0, 20}] (* Eric W. Weisstein, Mar 14 2026 *)
CoefficientList[Series[-(x^3 (1 + 3 x + 6 x^2 + 5 x^3)/(-1 + x)^7), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 14 2026 *)

Crossrefs

Cf. A382084.

Sequence in context: A341989 A341070 A373733 * A244871 A162617 A341139

Adjacent sequences: A382078 A382079 A382080 * A382082 A382083 A382084

Keyword

nonn,easy

Author

Enrique Navarrete, Mar 15 2025

Status

approved