A224289
Number of permutations of length n containing exactly 1 occurrence of 123 and 2 occurrences of 132.
Original entry on oeis.org
0, 0, 0, 2, 8, 26, 79, 232, 664, 1856, 5072, 13568, 35584, 91648, 232192, 579584, 1427456, 3473408, 8359936, 19922944, 47054848, 110231552, 256311296, 591921152, 1358430208, 3099590656, 7034896384, 15888023552, 35718692864, 79960211456, 178291474432, 396076515328, 876844417024
Offset: 1
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# Programs can be obtained from author's personal website.
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LinearRecurrence[{8,-24,32,-16},{0,0,0,2,8,26,79},40] (* Harvey P. Dale, Jun 23 2017 *)
A224291
Number of permutations of length n containing exactly 4 occurrences of 123 and 4 occurrences of 132.
Original entry on oeis.org
0, 0, 0, 0, 1, 11, 60, 270, 1084, 4028, 14144, 47577, 154740, 489728, 1514786, 4593118, 13682374, 40106060, 115824376, 329901232, 927585696, 2576685888, 7076644480, 19228648192, 51725149184
Offset: 1
A272099
Triangle read by rows, T(n,k) = C(n+1,k+1)*F([k-n, k-n-1], [-n-1], -1), where F is the generalized hypergeometric function, for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 4, 1, 12, 5, 1, 32, 18, 6, 1, 80, 56, 25, 7, 1, 192, 160, 88, 33, 8, 1, 448, 432, 280, 129, 42, 9, 1, 1024, 1120, 832, 450, 180, 52, 10, 1, 2304, 2816, 2352, 1452, 681, 242, 63, 11, 1, 5120, 6912, 6400, 4424, 2364, 985, 316, 75, 12, 1
Offset: 0
Triangle starts:
1;
4, 1;
12, 5, 1;
32, 18, 6, 1;
80, 56, 25, 7, 1;
192, 160, 88, 33, 8, 1;
448, 432, 280, 129, 42, 9, 1;
1024, 1120, 832, 450, 180, 52, 10, 1;
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T := (n,k) -> binomial(n+1,k+1)*hypergeom([k-n, k-n-1], [-n-1], -1):
seq(seq(simplify(T(n,k)),k=0..n),n=0..9);
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T[n_, k_] := Binomial[n+1, k+1] HypergeometricPFQ[{k-n, k-n-1}, {-n-1}, -1];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 22 2019 *)
A387185
a(n) = n*2^(n-1) + binomial(n,2)*2^(n-2) + binomial(n,3)*2^(n-3).
Original entry on oeis.org
0, 1, 5, 19, 64, 200, 592, 1680, 4608, 12288, 32000, 81664, 204800, 505856, 1232896, 2969600, 7077888, 16711680, 39124992, 90898432, 209715200, 480772096, 1095761920, 2484076544, 5603590144, 12582912000, 28135391232, 62662901760, 139049566208, 307492814848, 677799526400
Offset: 0
a(3) = 19 since the words are (number of permutations in parentheses): add (3), ade (6), aee (3), bbd (3), bbe (3), ccc (1).
a(4) = 64 since from the 81 strings of length 4 we subtract the following 17 (number of permutations in parentheses): 0000 (1), 1111 (1), 1112 (4), 1122 (6), 1222 (4), 2222 (1).
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a[n_] := Sum[2^(n-k)*Binomial[n, k], {k, 1, 3}]; Array[a, 30, 0] (* Amiram Eldar, Aug 21 2025 *)
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