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A262035 Number of permutations of [2n+1] beginning with exactly n ascents.

%I A262035 #6 Sep 09 2015 12:09:13
%S A262035 1,2,15,168,2520,47520,1081080,28828800,882161280,30474662400,
%T A262035 1173274502400,49819040409600,2313026876160000,116576554558464000,
%U A262035 6338850154116480000,369890550169620480000,23056510960573009920000,1529010726859052236800000
%N A262035 Number of permutations of [2n+1] beginning with exactly n ascents.
%H A262035 Alois P. Heinz, <a href="/A262035/b262035.txt">Table of n, a(n) for n = 0..365</a>
%F A262035 E.g.f.: (1-2*x)/(4*sqrt(1-4*x)*x^2)+(2*x^2-1)/(4*x^2).
%F A262035 a(n) = 2*(n+1)*(2*n+1)*a(n-1)/(n+2) for n>1, a(n) = n+1 for n<=1.
%F A262035 a(n) = (2*n+1)!/(n+1)! - floor((2*n+2)!/((n+1)!*(n+1)*(n+2)*2)).
%F A262035 a(n) = A262033(2n+1) - A262034(2n+1).
%e A262035 a(0) = 1: 1.
%e A262035 a(1) = 2: 132, 231.
%e A262035 a(2) = 15: 12435, 12534, 12543, 13425, 13524, 13542, 14523, 14532, 23415, 23514, 23541, 24513, 24531, 34512, 34521.
%p A262035 a:= proc(n) option remember; `if`(n<2, n+1,
%p A262035       2*(n+1)*(2*n+1)*a(n-1)/(n+2))
%p A262035     end:
%p A262035 seq(a(n), n=0..20);
%Y A262035 Cf. A262033, A262034.
%K A262035 nonn
%O A262035 0,2
%A A262035 _Alois P. Heinz_, Sep 08 2015