3,2

Lunnon's DE(n,n-2); Lunnon's DE(n,n-1) is number of free trees.

W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-67.

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^2(x^2)/(1-B(x^2)) where B(x) is generating function for rooted trees with n nodes

1 tromino in 1-space; 4 nonstraight tetrominoes in 2-space;

11 nonflat pentominoes in 3-space (chiral pairs count as one)

sb[ n_, k_ ] := sb[ n, k ]=b[ n+1-k, 1 ]+If[ n<2k, 0, sb[ n-k, k ] ]; b[ 1, 1 ] := 1;

b[ n_, 1 ] := b[ n, 1 ]=Sum[ b[ i, 1 ]sb[ n-1, i ]i, {i, 1, n-1} ]/(n-1);

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

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

+If[ OddQ[ i/2 ], 0, b[ i/4, 1 ]/4 ]+Sum[ b[ i/2, j ], {j, 3, i/2} ] ]

+Sum[ b[ j, 1 ](b[ i-2j, 1 ]/2+b[ i-2j, 2 ]/4)+Sum[ If[ OddQ[ k ], b[ j,

(k-1)/2 ]b[ i-2j, 1 ], 0 ], {k, 5, i} ], {j, 1, (i-1)/2} ], {i, 3, 30} ]

Cf. A000081, A000055, A036365.

Sequence in context: A104500 A119716 A000626 this_sequence A052550 A114248 A054577

Adjacent sequences: A036361 A036362 A036363 this_sequence A036365 A036366 A036367

easy,nice,nonn

Robert A. Russell (russell(AT)post.harvard.edu)