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Number of vectors of length n that are symmetric about the middle, where each element is drawn from a set of n distinct elements.
6

%I #39 Jun 25 2023 18:20:57

%S 1,1,2,9,16,125,216,2401,4096,59049,100000,1771561,2985984,62748517,

%T 105413504,2562890625,4294967296,118587876497,198359290368,

%U 6131066257801,10240000000000,350277500542221,584318301411328,21914624432020321,36520347436056576

%N Number of vectors of length n that are symmetric about the middle, where each element is drawn from a set of n distinct elements.

%H Alois P. Heinz, <a href="/A078707/b078707.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = n^(floor((n+1)/2)) = n^ceiling(n/2).

%e Examples added by _N. J. A. Sloane_, Jun 17 2014:

%e n=1: 1 (1).

%e n=2: 11, 22 (2).

%e n=3: 111X3, 121X6 (9).

%e n=4: 1111X4, 1221X12 (16).

%e n=5: 11111X5, 11211X20, 12221X20, 12121X20, 12321X60 (125).

%p a:= n-> n^ceil(n/2): seq(a(n), n=0..30); # _Alois P. Heinz_, Jul 23 2014

%t Join[{1}, Table[n^Ceiling[n/2], {n, 30}]] (* _Wesley Ivan Hurt_, Jan 15 2017 *)

%o (PARI) for(n=1,22,print1(n^((n+n%2)/2),","))

%Y Cf. A004526, A243520.

%Y Cf. A168658, A275549.

%Y This is for Coxeter type B what A152291 is for Coxeter type A.

%K nonn,easy

%O 0,3

%A _Mark Sterling_, Dec 18 2002

%E Extended by _Klaus Brockhaus_, Dec 19 2002

%E a(0)=1 inserted by _Alois P. Heinz_, Jul 23 2014