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 A289489 Number of permutations p of [n] such that in 0p the sum of all jumps equals 2n. 2
 1, 0, 0, 1, 4, 15, 104, 644, 3696, 23388, 151842, 979110, 6445659, 43148963, 290832906, 1977914328, 13574296048, 93787977144, 651970844448, 4558718881927, 32038664402074, 226200869873851, 1603811085640698, 11415385190127413, 81538284501095235 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..300 FORMULA a(n) = A291722(n,n). a(n) ~ c * d^n / n^2, where d = 7.7572369635460295... and c = 0.022080578979754... - Vaclav Kotesovec, Nov 17 2022 EXAMPLE a(3) = 1: 312. a(4) = 4: 3142, 4213, 4231, 4312. a(5) = 15: 15234, 25134, 31542, 35124, 41235, 42153, 42531, 43152, 45123, 53214, 53241, 53421, 54213, 54231, 54312. a(6) = 104: 126354, 136254, 142635, 146253, ..., 653421, 654213, 654231, 654312. MAPLE b:= proc(u, o) option remember; expand(`if`(u+o=0, 1, add(b(u-j, o+j-1)*x^(j-1), j=1..u)+ add(b(u+j-1, o-j)*x^(j-1), j=1..o))) end: a:= n-> coeff(b(0, n), x, n): seq(a(n), n=0..26); MATHEMATICA b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1, Sum[b[u - j, o + j - 1]*x^(j - 1), {j, 1, u}] + Sum[b[u + j - 1, o - j]*x^(j - 1), {j, 1, o}]]]; a[n_] := Coefficient[b[0, n], x, n]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Nov 17 2022, after Alois P. Heinz *) CROSSREFS Cf. A291722. Sequence in context: A352415 A079128 A356524 * A221095 A339287 A081548 Adjacent sequences: A289486 A289487 A289488 * A289490 A289491 A289492 KEYWORD nonn AUTHOR Alois P. Heinz, Sep 02 2017 STATUS approved

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Last modified June 15 03:12 EDT 2024. Contains 373402 sequences. (Running on oeis4.)