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 A302691 Number of broken alternating permutations of n things. 2
 0, 0, 1, 2, 7, 26, 117, 594, 3407, 21682, 151853, 1160026, 9600567, 85566378, 817099909, 8322907298, 90074979487, 1032183177314, 12485056392285, 158964674218410, 2125201153260167, 29764791617545690, 435823661971532981, 6658895050949717362, 105979606291488794607 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A permutation of {1,2,...,n} is said to be a "broken alternating permutation" if it is an alternating permutation (cf. A000111) except at one point. See El Hilany and Rau for precise definition and an explicit formula. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..484 D. Chebikin, Variations on descents and inversions in permutations, The Electronic J. of Combinatorics, 15 (2008), #R132. Boulos El Hilany, Johannes Rau, Signed counts of real simple rational functions, arXiv:1712.05639 [math.AG], 2017, Proposition 6.4, p. 19. FORMULA a(n) ~ (4 - Pi) * 2^(n + 5/2) * n^(n + 3/2) / (exp(n) * Pi^(n + 3/2)). - Vaclav Kotesovec, Apr 14 2018 E.g.f.: (cos(x)-sin(x)+x-1)/(sin(x)-1). - Alois P. Heinz, Apr 16 2018 MAPLE b:= proc(u, o, t) option remember; `if`(u+o=0, t, add(b(o+j-1, u-j, t), j=1..u)+ `if`(t=0, add(b(o-j, u-1+j, 1), j=1..o), 0)) end: a:= n-> b(n, 0\$2): seq(a(n), n=0..25); # Alois P. Heinz, Apr 14 2018 # second Maple program: egf:= (cos(x)-sin(x)+x-1)/(sin(x)-1): a:= n-> n! * coeff(series(egf, x, n+1), x, n): seq(a(n), n=0..25); # Alois P. Heinz, Apr 16 2018 MATHEMATICA b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, t, Sum[b[o + j - 1, u - j, t], {j, 1, u}] + If[t == 0, Sum[b[o - j, u - 1 + j, 1], {j, 1, o}], 0]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 19 2018, after Alois P. Heinz *) CROSSREFS Cf. A000111. Column k=2 of A145876. Sequence in context: A167551 A309396 A218670 * A081566 A213094 A141203 Adjacent sequences: A302688 A302689 A302690 * A302692 A302693 A302694 KEYWORD nonn AUTHOR Michael De Vlieger, Apr 11 2018 EXTENSIONS a(13)-a(24) from Alois P. Heinz, Apr 14 2018 STATUS approved

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Last modified June 4 10:24 EDT 2023. Contains 363121 sequences. (Running on oeis4.)