A322295 Number of permutations of [2n] with exactly n rising or falling successions.

1, 2, 10, 120, 2198, 54304, 1674468, 61736880, 2644978110, 129019925424, 7056278570108, 427516982398576, 28417031969575260, 2055803302988520320, 160786792285272198088, 13517797622576903566560, 1215656330296317906395790, 116440107306323155369822800

Offset: 0

Author

Alois P. Heinz, Dec 02 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..348

Crossrefs

Bisection (even part) of A322294.

Cf. A001100.

Programs

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), t, n):
seq(a(n), n=0..20);

Mathematica

s[n_] := s[n] = If[n < 4, {1, 1, 2t, 4t + 2 t^2}[[n+1]], Expand[(n+1-t)* s[n-1] - (1-t)(n-2 + 3t) s[n-2] - (1-t)^2 (n-5+t) s[n-3] + (1-t)^3 (n-3)* s[n-4]]];
a[n_] := Coefficient[s[2n], t, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 23 2021, after Alois P. Heinz *)

Formula

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