A384243
a(n) = 2^(n-6)*n*(n^3 - 6*n^2 + 19*n - 14).
Original entry on oeis.org
0, 0, 1, 6, 30, 140, 600, 2352, 8512, 28800, 92160, 281600, 827904, 2356224, 6522880, 17633280, 46694400, 121438208, 310837248, 784465920, 1954938880, 4816896000, 11747721216, 28386000896, 68010639360, 161690419200, 381681664000, 895098028032, 2086448136192, 4836200284160
Offset: 0
a(4) = 30 since the strings are the 6 permutations of 2233, the 6 permutations of 1122, the 6 permutations of 0022, and the 12 permutations of 0122.
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A384243[n_] := 2^(n-6)*n*(n-1)*(n*(n-5)+14); Array[A384243, 30, 0] (* or *)
LinearRecurrence[{10, -40, 80, -80, 32}, {0, 0, 1, 6, 30}, 30] (* Paolo Xausa, May 27 2025 *)
A384506
a(n) = 2^(n-7)*(n^4 - 6*n^3 + 59*n^2 - 54*n)/3.
Original entry on oeis.org
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
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.
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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 *)
A383868
a(n) = 2^(n-3)*(3*binomial(n,4) + 4*binomial(n,2) + 8).
Original entry on oeis.org
1, 2, 6, 20, 70, 252, 904, 3152, 10560, 33920, 104704, 311808, 899584, 2524160, 6912000, 18526208, 48726016, 126025728, 321126400, 807403520, 2005794816, 4929093632, 11994136576, 28924968960, 69185044480, 164240556032, 387201368064, 907009851392, 2112083722240
Offset: 0
a(2) = 6 since the strings are 01, 10, 00, 11, 22, and 33.
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a[n_] := 2^(n-3) * (3*Binomial[n, 4] + 4*Binomial[n, 2] + 8); Array[a, 30, 0] (* Amiram Eldar, May 13 2025 *)
Showing 1-3 of 3 results.
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