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A362581 Number of alternating permutations on [2n+1] with 1 in position n+1.

%I A362581 #17 Apr 26 2023 07:46:07
%S A362581 1,2,6,80,1750,64512,3438204,253913088,24687555750,3062092267520,
%T A362581 471565937953396,88298062293762048,19753693667117055100,
%U A362581 5203824518733863321600,1594426273578194363292600,562191171748426920367226880,226024705816530632892282399750
%N A362581 Number of alternating permutations on [2n+1] with 1 in position n+1.
%H A362581 Alois P. Heinz, <a href="/A362581/b362581.txt">Table of n, a(n) for n = 0..242</a>
%F A362581 a(n) = binomial(2*n,n)*A000111(n)^2:
%F A362581 a(n) = A104345(2*n,n).
%e A362581 a(0) = 1: 1.
%e A362581 a(1) = 2: 213, 312.
%e A362581 a(2) = 6: 23154, 24153, 25143, 34152, 35142, 45132.
%p A362581 b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p A362581       add(b(o-1+j, u-j), j=1..u))
%p A362581     end:
%p A362581 a:= n-> binomial(2*n, n)*b(n, 0)^2:
%p A362581 seq(a(n), n=0..20);
%o A362581 (Python)
%o A362581 from itertools import accumulate
%o A362581 from math import comb
%o A362581 def A362581(n):
%o A362581     if n <= 1: return n+1
%o A362581     blist = (0,1)
%o A362581     for _ in range(n-1):
%o A362581         blist = tuple(accumulate(reversed(blist),initial=0))
%o A362581     return blist[-1]**2*comb(n<<1,n) # _Chai Wah Wu_, Apr 25 2023
%Y A362581 Cf. A000111, A104345.
%K A362581 nonn
%O A362581 0,2
%A A362581 _Alois P. Heinz_, Apr 25 2023