A384506 a(n) = 2^(n-7)*(n^4 - 6*n^3 + 59*n^2 - 54*n)/3.

0, 0, 1, 6, 25, 90, 300, 952, 2912, 8640, 24960, 70400, 194304, 525824, 1397760, 3655680, 9420800, 23953408, 60162048, 149422080, 367329280, 894566400, 2159804416, 5173149696, 12299796480, 29045555200, 68157440000, 158997676032, 368880648192, 851443712000, 1955887841280

Offset

0,4

Comments

a(n) is the number of strings of length n defined on {0, 1, 2, 3} that have exactly two 2's and no 3's or exactly four 3's and no 2's.

Links

Table of n, a(n) for n=0..30.

Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Formula

E.g.f.: exp(2*x)*(x^2/2 + x^4/24).

G.f.: x^2*(1 - 4*x + 5*x^2)/(1 - 2*x)^5. - Stefano Spezia, Jun 01 2025

Example

a(4) = 25 since the strings are the 6 permutations of 2200, the 6 permutations of 2211, the 12 permutations of 2201, and 3333.
a(6) = 300 since the strings are (number of permutations in parentheses): 220000 (15), 220001 (60), 220011 (90), 220111 (60), 221111 (15), 333300 (15), 333301 (30), and 333311 (15). Note that the 15 permutations of the string 223333 are excluded.

Mathematica

CoefficientList[Series[x^2*(1 - 4*x + 5*x^2)/(1 - 2*x)^5, {x, 0, 30}], x] (* or *) LinearRecurrence[{10, -40, 80, -80, 32}, {0, 0, 1, 6, 25}, 31] (* James C. McMahon, Jun 04 2025 *)

Crossrefs

Cf. A383778, A384243.

Sequence in context: A056279 A055337 A309946 * A001871 A000392 A365531

Adjacent sequences: A384503 A384504 A384505 * A384507 A384508 A384509

Keyword

nonn,easy

Author

Enrique Navarrete, May 31 2025

Status

approved