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A036367 Number of free orthoplex n-ominoes with cell centers determining n-2 space.

%I A036367 #15 Feb 06 2021 22:28:42
%S A036367 1,2,8,25,86,272,875,2732,8505,26104,79708,241522,728632,2187951,
%T A036367 6548819,19542662,58184124,172880565,512837063,1519158462,4494920802,
%U A036367 13286473612,39240530012,115811180864,341588823740,1007007175952,2967361180383
%N A036367 Number of free orthoplex n-ominoes with cell centers determining n-2 space.
%C A036367 Orthoplex polyominoes are multidimensional polyominoes that do not extend more than two units along any axis.
%F A036367 G.f.: (B^2(x) + B(x^2))^2/8 + B^2(x^2)/4 + B(x^4)/4 + B^5(x)/(2 - 2*B(x)) + (B(x) + B(x^2))*B^2(x^2)/(2 - 2*B(x^2)) where B(x) is the generating function for rooted trees with n nodes in A000081.
%e A036367 a(4)=1 because there is 1 tetromino (a square) in 2^2 space;
%e A036367 a(5)=2 because there are 2 pentominoes in 2^3 space;
%e A036367 a(6)=8 because in 2^4 space there are 8 hexominoes that have cell centers determining 4-space.
%t A036367 sb[ n_, k_ ] := sb[ n, k ]=b[ n+1-k, 1 ]+If[ n<2k, 0, sb[ n-k, k ] ]; b[ 1, 1 ] := 1;
%t A036367 b[ n_, 1 ] := b[ n, 1 ]=Sum[ b[ i, 1 ]sb[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
%t A036367 b[ n_, k_ ] := b[ n, k ]=Sum[ b[ i, 1 ]b[ n-i, k-1 ], {i, 1, n-1} ];
%t A036367 Table[ b[ i, 4 ]/8+Sum[ b[ i, j ], {j, 5, i} ]/2+If[ OddQ[ i ], 0, 3b[ i/2, 2 ]/8
%t A036367 +If[ OddQ[ i/2 ], 0, b[ i/4, 1 ]/4 ]+Sum[ b[ i/2, j ], {j, 3, i/2} ]/2 ]
%t A036367 +Sum[ b[ j, 1 ]b[ i-2j, 2 ]/4+Sum[ If[ OddQ[ k ], b[ j,
%t A036367 (k-1)/2 ]b[ i-2j, 1 ], 0 ], {k, 5, i} ]/2, {j, 1, (i-1)/2} ], {i, 4, 30} ]
%Y A036367 Cf. A000081, A036364.
%K A036367 easy,nice,nonn
%O A036367 4,2
%A A036367 _Robert A. Russell_