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%I #28 Apr 05 2022 15:53:22
%S 1,4,11,35,104,319,951,2862,8516,25369,75167,222529,656961,1937393,
%T 5704426,16781247,49320800,144866243,425263010,1247877578,3660478408,
%U 10734834603,31475111515,92273758477,270486112046,792836030163,2323835125879,6811162237825
%N Number of free n-ominoes with cell centers determining n-2 space (proper dimension n-2).
%C Lunnon's DE(n,n-2); Lunnon's DE(n,n-1) is number of free trees.
%H Vaclav Kotesovec, <a href="/A036364/b036364.txt">Table of n, a(n) for n = 3..1000</a> (terms 3..200 from Vincenzo Librandi)
%H 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 G.f.: B^3(x)/2 + B(x)B(x^2)/2 + 5B^4(x)/8 + B^2(x)B(x^2)/4 + 7B^2(x^2)/8 + B(x^4)/4 + B^5(x)/(1-B(x)) + (B(x)+B(x^2))B^2(x^2)/(1-B(x^2)), where B(x) is the generating function for rooted trees with n nodes (that is, B(x) is the g.f. of sequence A000081).
%F a(n) ~ A340310 * A051491^n / sqrt(n). - _Vaclav Kotesovec_, Apr 12 2021
%e 1 tromino in 1-space;
%e 4 nonstraight tetrominoes in 2-space;
%e 11 nonflat pentominoes in 3-space (chiral pairs count as one).
%t sb[ n_, k_ ] := sb[ n, k ]=b[ n+1-k, 1 ]+If[ n<2k, 0, sb[ n-k, k ] ]; b[ 1, 1 ] := 1;
%t b[ n_, 1 ] := b[ n, 1 ]=Sum[ b[ i, 1 ]sb[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
%t b[ n_, k_ ] := b[ n, k ]=Sum[ b[ i, 1 ]b[ n-i, k-1 ], {i, 1, n-1} ];
%t Table[ b[ i, 3 ]/2+5b[ i, 4 ]/8+Sum[ b[ i, j ], {j, 5, i} ]+If[ OddQ[ i ], 0, 7b[ i/2, 2 ]/8
%t +If[ OddQ[ i/2 ], 0, b[ i/4, 1 ]/4 ]+Sum[ b[ i/2, j ], {j, 3, i/2} ] ]
%t +Sum[ b[ j, 1 ](b[ i-2j, 1 ]/2+b[ i-2j, 2 ]/4)+Sum[ If[ OddQ[ k ], b[ j,
%t (k-1)/2 ]b[ i-2j, 1 ], 0 ], {k, 5, i} ], {j, 1, (i-1)/2} ], {i, 3, 30} ]
%Y Cf. A000081, A000055, A036365, A171860 (fixed).
%K easy,nice,nonn
%O 3,2
%A _Robert A. Russell_
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