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

%I A322296 #12 Apr 21 2022 05:22:25
%S A322296 1,4,48,888,22120,685368,25344480,1087931184,53138966904,
%T A322296 2909014993080,176372774697856,11729862804913680,848948339328178128,
%U A322296 66420006805308507568,5585680154203107163200,502437191145813112268640,48134705092961286591532440
%N A322296 Number of permutations of [2n+1] with exactly n rising or falling successions.
%H A322296 Alois P. Heinz, <a href="/A322296/b322296.txt">Table of n, a(n) for n = 0..347</a>
%F A322296 a(n) = A001100(2n+1,n).
%p A322296 S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
%p A322296        [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)
%p A322296        -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))
%p A322296     end:
%p A322296 a:= n-> coeff(S(2*n+1), t, n):
%p A322296 seq(a(n), n=0..20);
%t A322296 S[n_] := S[n] = If[n < 4, {1, 1, 2*t, 4*t + 2*t^2}[[n + 1]], Expand[
%t A322296                (n + 1 - t)*S[n - 1] -
%t A322296      (1 - t)*(n - 2 + 3*t)*S[n - 2] -
%t A322296      (1 - t)^2*(n - 5 + t)*S[n - 3] +
%t A322296          (1 - t)^3*(n - 3)*S[n - 4]]];
%t A322296 a[n_] := Coefficient[S[2*n + 1], t, n];
%t A322296 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 21 2022, after _Alois P. Heinz_ *)
%Y A322296 Bisection (odd part) of A322294.
%Y A322296 Cf. A001100.
%K A322296 nonn
%O A322296 0,2
%A A322296 _Alois P. Heinz_, Dec 02 2018