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A036366 Number of asymmetric n-ominoes in n-2 space.

%I A036366 #20 Jan 13 2019 03:17:44
%S A036366 0,1,4,13,42,113,309,792,2049,5167,13071,32724,82006,204619,510655,
%T A036366 1272101,3168971,7888446,19636642,48868367,121621466,302673515,
%U A036366 753319709,1875049668,4667676111,11620911254,28936281066,72062264255
%N A036366 Number of asymmetric n-ominoes in n-2 space.
%H A036366 W. F. Lunnon, <a href="https://doi.org/10.1093/comjnl/18.4.366">Counting Multidimensional Polyominoes</a>, Computer Journal, Vol. 18 (1975), pp. 366-67.
%F A036366 G.f.: A^3(x)/2 - A(x)A(x^2)/2 + 5A^4(x)/8 - A^2(x)A(x^2)/4 - 5A^2(x^2)/8 + A(x^4)/4 + A^5(x)/(1-A(x)) - (A(x)+A(x^2))*A^2(x^2)/(1-A(x^2)), where A(x) is the generating function for rooted identity trees with n nodes (that is, the g.f. of sequence A004111).
%e A036366 0 asymmetric trominoes in 1-space;
%e A036366 1 asymmetric tetromino in 2-space;
%e A036366 4 asymmetric pentominoes in 3-space.
%t A036366 sa[ n_, k_ ] := sa[ n, k ]=a[ n+1-k, 1 ]+If[ n<2k, 0, -sa[ n-k, k ] ]; a[ 1, 1 ] := 1;
%t A036366 a[ n_, 1 ] := a[ n, 1 ]=Sum[ a[ i, 1 ]sa[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
%t A036366 a[ n_, k_ ] := a[ n, k ]=Sum[ a[ i, 1 ]a[ n-i, k-1 ], {i, 1, n-1} ];
%t A036366 Table[ a[ i, 3 ]/2+5a[ i, 4 ]/8+Sum[ a[ i, j ], {j, 5, i} ]-If[ OddQ[ i ], 0, 5a[ i/2, 2 ]/8
%t A036366 -If[ OddQ[ i/2 ], 0, a[ i/4, 1 ]/4 ]+Sum[ a[ i/2, j ], {j, 3, i/2} ] ]
%t A036366 -Sum[ a[ j, 1 ](a[ i-2j, 1 ]/2+a[ i-2j, 2 ]/4)+Sum[ If[ OddQ[ k ], a[ j,
%t A036366 (k-1)/2 ]a[ i-2j, 1 ], 0 ], {k, 5, i} ], {j, 1, (i-1)/2} ], {i, 3, 30} ]
%Y A036366 Cf. A004111, A000220, A036365.
%K A036366 easy,nice,nonn
%O A036366 3,3
%A A036366 _Robert A. Russell_