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A317281 Number of permutations of [n] with exactly one increasing run of even length.

%I A317281 #8 Jul 01 2019 18:21:40
%S A317281 1,4,12,52,299,1750,11195,83074,675304,5880354,55691345,570430818,
%T A317281 6235219145,72578854088,898625380692,11776375141808,162751388533495,
%U A317281 2367374376626798,36156236738459671,578320016708416430,9668632218886209536,168657264917156460894
%N A317281 Number of permutations of [n] with exactly one increasing run of even length.
%H A317281 Alois P. Heinz, <a href="/A317281/b317281.txt">Table of n, a(n) for n = 2..473</a>
%F A317281 a(n) ~ c * d^n * n^(n + 3/2), where d = 0.26695170015188549567... and c = 1.1281954524592228... - _Vaclav Kotesovec_, Jul 01 2019
%p A317281 b:= proc(u, o, t) option remember; series(`if`(u+o=0, x^t,
%p A317281       add(b(u+j-1, o-j, irem(t+1, 2)), j=1..o)+
%p A317281       add(b(u-j, o+j-1, 0)*x^t, j=1..u)), x, 2)
%p A317281     end:
%p A317281 a:= n-> coeff(b(n, 0$2), x, 1):
%p A317281 seq(a(n), n=2..25);
%Y A317281 Column k=1 of A097592.
%K A317281 nonn
%O A317281 2,2
%A A317281 _Alois P. Heinz_, Jul 25 2018