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A262745 Number of permutations of [n] with an odd number of rises.

%I A262745 #11 Nov 01 2021 04:23:42
%S A262745 0,0,1,4,12,52,360,2656,20160,177472,1814400,20135296,239500800,
%T A262745 3102326272,43589145600,654789062656,10461394944000,177738781376512,
%U A262745 3201186852864000,60837094646972416,1216451004088320000,25542995336828157952,562000363888803840000
%N A262745 Number of permutations of [n] with an odd number of rises.
%H A262745 Alois P. Heinz, <a href="/A262745/b262745.txt">Table of n, a(n) for n = 0..400</a>
%F A262745 E.g.f.: (1/(1-x) -1 -tanh(x))/2.
%F A262745 a(n) = A000142(n) - A128103(n).
%e A262745 a(2) = 1: 12.
%e A262745 a(3) = 4: 132, 213, 231, 312.
%p A262745 b:= proc(u, o, t) option remember; `if`(u+o=0, t,
%p A262745       add(b(u-j, o+j-1, t), j=1..u)+
%p A262745       add(b(u+j-1, o-j, 1-t), j=1..o))
%p A262745     end:
%p A262745 a:= n-> b(n, 0$2):
%p A262745 seq(a(n), n=0..25);
%t A262745 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, t,
%t A262745      Sum[b[u - j, o + j - 1, t], {j, 1, u}] +
%t A262745      Sum[b[u + j - 1, o - j, 1 - t], {j, 1, o}]];
%t A262745 a[n_] := b[n, 0, 0];
%t A262745 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Nov 01 2021, after _Alois P. Heinz_ *)
%Y A262745 Cf. A000142, A128103.
%K A262745 nonn
%O A262745 0,4
%A A262745 _Alois P. Heinz_, Sep 29 2015