%I #13 Feb 06 2021 22:29:20

%S 0,0,4,14,37,110,324,888,2368,6336,16874,44414,116181,303362,790157,

%T 2051880,5317599,13764133,35586766,91910082,237183164,611701614,

%U 1576773162,4062606255,10463699696,26942811809,69358469092

%N Number of chiral orthoplex n-ominoes in n-2 space.

%C Orthoplex polyominoes are multidimensional polyominoes that do not extend more than two units along any axis.

%F G.f.: (C^2(x) + C(-x^2))^2/8 - C^2(-x^2)/4 - C(-x^4)/4 + C^5(x)/(2-2C(x)) - (C(x)+C(-x^2))*C^2(-x^2)/(2-2C(-x^2)) where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

%e a(6)=4 because there are 4 pairs of chiral hexominoes in 2^4 space.

%t sc[ n_, k_ ] := sc[ n, k ]=c[ n+1-k, 1 ]+If[ n<2k, 0, sc[ n-k, k ](-1)^k ]; c[ 1, 1 ] := 1;

%t c[ n_, 1 ] := c[ n, 1 ]=Sum[ c[ i, 1 ]sc[ n-1, i ]i, {i, 1, n-1} ]/(n-1);

%t c[ n_, k_ ] := c[ n, k ]=Sum[ c[ i, 1 ]c[ n-i, k-1 ], {i, 1, n-1} ];

%t Table[ c[ i, 4 ]/8+Sum[ c[ i, j ], {j, 5, i} ]/2-If[ OddQ[ i ], 0,

%t c[ i/2, 2 ](-1)^(i/2)/8+If[ OddQ[ i/2 ], 0, c[ i/4, 1 ](-1)^(i/4)/4 ]

%t +Sum[ c[ i/2, j ](-1)^(i/2), {j, 3, i/2} ]/2 ]+Sum[ c[ j, 1 ]c[ i-2j, 2 ](-1)^j/4

%t -Sum[ If[ OddQ[ k ], c[ j, (k-1)/2 ]c[ i-2j, 1 ](-1)^j/2, 0 ], {k, 5, i} ],

%t {j, 1, (i-1)/2} ], {i, 4, 30} ]

%Y Cf. A045648, A036367.

%K easy,nice,nonn

%O 4,3

%A _Robert A. Russell_