This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384506 #19 Jun 04 2025 11:07:27
%S A384506 0,0,1,6,25,90,300,952,2912,8640,24960,70400,194304,525824,1397760,
%T A384506 3655680,9420800,23953408,60162048,149422080,367329280,894566400,
%U A384506 2159804416,5173149696,12299796480,29045555200,68157440000,158997676032,368880648192,851443712000,1955887841280
%N A384506 a(n) = 2^(n-7)*(n^4 - 6*n^3 + 59*n^2 - 54*n)/3.
%C A384506 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.
%H A384506 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).
%F A384506 E.g.f.: exp(2*x)*(x^2/2 + x^4/24).
%F A384506 G.f.: x^2*(1 - 4*x + 5*x^2)/(1 - 2*x)^5. - _Stefano Spezia_, Jun 01 2025
%e A384506 a(4) = 25 since the strings are the 6 permutations of 2200, the 6 permutations of 2211, the 12 permutations of 2201, and 3333.
%e A384506 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.
%t A384506 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 *)
%Y A384506 Cf. A383778, A384243.
%K A384506 nonn,easy
%O A384506 0,4
%A A384506 _Enrique Navarrete_, May 31 2025