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%I #13 Feb 06 2021 22:29:59
%S 0,0,2,10,33,101,277,754,1972,5134,13145,33563,85048,215075,541887,
%T 1363376,3423537,8588521,21522243,53898911,134895022,337467395,
%U 843924068,2109897826,5273789335,13180026621,32934865984
%N Number of asymmetric 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.: (A^2(x)-A(x^2))^2/8 - A^2(x^2)/4 + A(x^4)/4 + A^5(x)/(2-2A(x)) - (A(x)+A(x^2))*A^2(x^2)/(2-2A(x^2)) where A(x) is the generating function for rooted identity trees with n nodes in A004111.
%e a(6)=4 because there are 2 asymmetric hexominoes in 2^4 space.
%t sa[ n_, k_ ] := sa[ n, k ]=a[ n+1-k, 1 ]+If[ n<2k, 0, -sa[ n-k, k ] ]; a[ 1, 1 ] := 1;
%t a[ n_, 1 ] := a[ n, 1 ]=Sum[ a[ i, 1 ]sa[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
%t a[ n_, k_ ] := a[ n, k ]=Sum[ a[ i, 1 ]a[ n-i, k-1 ], {i, 1, n-1} ];
%t Table[ a[ i, 4 ]/8+Sum[ a[ i, j ], {j, 5, i} ]/2-If[ OddQ[ i ], 0, a[ i/2, 2 ]/8
%t -If[ OddQ[ i/2 ], 0, a[ i/4, 1 ]/4 ]+Sum[ a[ i/2, j ], {j, 3, i/2} ]/2 ]
%t -Sum[ a[ j, 1 ]a[ i-2j, 2 ]/4+Sum[ If[ OddQ[ k ], a[ j,
%t (k-1)/2 ]a[ i-2j, 1 ], 0 ], {k, 5, i} ]/2, {j, 1, (i-1)/2} ], {i, 4, 30} ]
%Y Cf. A004111, A036366.
%K easy,nice,nonn
%O 4,3
%A _Robert A. Russell_
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