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A001268
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One-half the number of permutations of length n with exactly 4 rising or falling successions.
(Formerly M4805 N2053)
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5
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0, 0, 0, 0, 0, 1, 11, 113, 1099, 11060, 118484, 1366134, 16970322, 226574211, 3240161105, 49453685911, 802790789101, 13815657556958, 251309386257874, 4818622686395380, 97145520138758844, 2054507019515346789, 45484006970415223287, 1052036480881734378541
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OFFSET
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0,7
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COMMENTS
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(1/2) times number of permutations of 12...n such that exactly 4 of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
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REFERENCES
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F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
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).
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LINKS
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FORMULA
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Coefficient of t^4 in S[n](t) defined in A002464, divided by 2.
Recurrence (for n>5): (n-5)*(n^8 - 41*n^7 + 730*n^6 - 7358*n^5 + 45799*n^4 - 179702*n^3 + 432498*n^2 - 581244*n + 332100)*a(n) = (n^10 - 45*n^9 + 895*n^8 - 10301*n^7 + 75340*n^6 - 361190*n^5 + 1124682*n^4 - 2150033*n^3 + 2147364*n^2 - 499899*n - 544266)*a(n-1) - (n^10 - 44*n^9 + 869*n^8 - 10112*n^7 + 76390*n^6 - 388742*n^5 + 1336932*n^4 - 3028095*n^3 + 4237931*n^2 - 3198426*n + 917988)*a(n-2) - (n^10 - 43*n^9 + 823*n^8 - 9195*n^7 + 66108*n^6 - 318138*n^5 + 1033118*n^4 - 2224673*n^3 + 3023402*n^2 - 2325285*n + 761190)*a(n-3) + (n^8 - 33*n^7 + 471*n^6 - 3783*n^5 + 18594*n^4 - 56865*n^3 + 104723*n^2 - 104847*n + 42783)*(n-2)^2*a(n-4). - Vaclav Kotesovec, Aug 11 2013
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MAPLE
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S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
[n+1], expand((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)))
end:
a:= n-> ceil(coeff(S(n), t, 4)/2):
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MATHEMATICA
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S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t + 2*t^2}[[n+1]], Expand[(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]]]; a[n_] := Ceiling[Coefficient[S[n], t, 4]/2]; Table [a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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