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Number of permutations p of [n] such that p(i), p(i+2), p(i+4),... form an up-down sequence for i in {1,2}.
2

%I #28 Dec 06 2023 14:44:21

%S 1,1,2,3,6,20,80,350,1750,10080,64512,450912,3438204,28471872,

%T 253913088,2424193200,24687555750,267199961600,3062092267520,

%U 37037541651968,471565937953396,6304419553216512,88298062293762048,1292879475255280640,19753693667117055100

%N Number of permutations p of [n] such that p(i), p(i+2), p(i+4),... form an up-down sequence for i in {1,2}.

%C Number of permutations p of [n] such that p(i) < p(i+2) > p(i+4) < ... for i <= 2.

%H Alois P. Heinz, <a href="/A361648/b361648.txt">Table of n, a(n) for n = 0..485</a>

%F a(n) = A000111(floor(n/2))*A000111(ceiling(n/2))*A001405(n).

%e a(0) = 1: (), the empty permutation.

%e a(1) = 1: 1.

%e a(2) = 2: 12, 21.

%e a(3) = 3: 123, 132, 213.

%e a(4) = 6: 1234, 1243, 1324, 2134, 2143, 3142.

%e a(5) = 20: 12453, 12534, 12543, 13452, 13542, 14352, 21453, 21534, 21543, 23451, 23541, 24351, 31452, 31524, 31542, 32451, 32541, 41523, 41532, 42531.

%e a(6) = 80: 124635, 125634, 125643, 126453, ..., 526413, 526431, 536412, 536421.

%p b:= proc(u, o) option remember; `if`(u+o=0, 1,

%p add(b(o-1+j, u-j), j=1..u))

%p end:

%p a:= n-> (h-> b(h, 0)*b(n-h, 0)*binomial(n, h))(iquo(n, 2)):

%p seq(a(n), n=0..30);

%t b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];

%t a[n_] := With[{h = Quotient[n, 2]}, b[h, 0] b[n-h, 0] Binomial[n, h]];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 26 2023, after _Alois P. Heinz_ *)

%o (Python)

%o from math import comb

%o from itertools import accumulate

%o def A361648(n):

%o if n<=1:

%o return 1

%o blist = (0,1)

%o for _ in range((m:=n>>1)-1):

%o blist = tuple(accumulate(reversed(blist),initial=0))

%o return blist[-1]*sum(blist)*comb(n,m) if n&1 else blist[-1]**2*comb(n,m) # _Chai Wah Wu_, Apr 16 2023

%Y Column k=2 of A361651.

%Y Cf. A000111, A000142, A001405.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Mar 19 2023