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 A000239 One-half of number of permutations of [n] with exactly one run of adjacent symbols differing by 1. (Formerly M2758 N1109) 3
 1, 1, 3, 8, 28, 143, 933, 7150, 62310, 607445, 6545935, 77232740, 989893248, 13692587323, 203271723033, 3223180454138, 54362625941818, 971708196867905, 18347779304380995, 364911199401630640, 7624625589633857940, 166977535317365068775, 3824547112283439914893, 91440772473772839055238 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS First differences seem to be in A000130. - Ralf Stephan, Aug 28 2003 REFERENCES F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.2. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Jean-François Alcover, Table of n, a(n) for n = 1..40 EXAMPLE The permutation 3 2 1 4 5 7 6 has three such runs: 3-2-1, 4-5 and 7-6. For n<=3 all permutations have one such run. For n=4, 16 have one run, two have no such runs (2413 and 3142), and 6 have two runs (1243, 2134, 2143, 3412, 3421), so a(4) = 16/2 = 8. MATHEMATICA S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t + 2*t^2}[[n+1]], (n+1-t)* S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4]]; A000239 = Join[{1}, Table[Coefficient[S[n], t, 1]/2, {n, 1, 20}] // Accumulate // Rest] (* Jean-François Alcover, Feb 06 2016, from successive accumulation of A000130 *) CROSSREFS This is a diagonal of the irregular triangle in A010030. Sequence in context: A135583 A009437 A000776 * A268302 A195687 A060707 Adjacent sequences:  A000236 A000237 A000238 * A000240 A000241 A000242 KEYWORD nonn AUTHOR EXTENSIONS Entry revised by N. J. A. Sloane, Apr 14 2014 More terms from Jean-François Alcover, Feb 06 2016 STATUS approved

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