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

%I A322295 #13 Oct 23 2021 10:13:19
%S A322295 1,2,10,120,2198,54304,1674468,61736880,2644978110,129019925424,
%T A322295 7056278570108,427516982398576,28417031969575260,2055803302988520320,
%U A322295 160786792285272198088,13517797622576903566560,1215656330296317906395790,116440107306323155369822800
%N A322295 Number of permutations of [2n] with exactly n rising or falling successions.
%H A322295 Alois P. Heinz, <a href="/A322295/b322295.txt">Table of n, a(n) for n = 0..348</a>
%F A322295 a(n) = A001100(2n,n).
%p A322295 S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
%p A322295        [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)
%p A322295        -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))
%p A322295     end:
%p A322295 a:= n-> coeff(S(2*n), t, n):
%p A322295 seq(a(n), n=0..20);
%t A322295 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]]];
%t A322295 a[n_] := Coefficient[s[2n], t, n];
%t A322295 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Oct 23 2021, after _Alois P. Heinz_ *)
%Y A322295 Bisection (even part) of A322294.
%Y A322295 Cf. A001100.
%K A322295 nonn
%O A322295 0,2
%A A322295 _Alois P. Heinz_, Dec 02 2018