A322296 Number of permutations of [2n+1] with exactly n rising or falling successions.

1, 4, 48, 888, 22120, 685368, 25344480, 1087931184, 53138966904, 2909014993080, 176372774697856, 11729862804913680, 848948339328178128, 66420006805308507568, 5585680154203107163200, 502437191145813112268640, 48134705092961286591532440

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..347

Formula

a(n) = A001100(2n+1,n).

Maple

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-> coeff(S(2*n+1), t, n):
seq(a(n), n=0..20);

Mathematica

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_] := Coefficient[S[2*n + 1], t, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 21 2022, after Alois P. Heinz *)

Crossrefs

Bisection (odd part) of A322294.

Cf. A001100.

Sequence in context: A047711 A089448 A167141 * A360484 A192260 A162676

Adjacent sequences: A322293 A322294 A322295 * A322297 A322298 A322299

Keyword

nonn

Author

Alois P. Heinz, Dec 02 2018

Status

approved