A229915 Number of espalier polycubes of a given volume in dimension 3.
1, 1, 3, 5, 10, 14, 26, 34, 57, 76, 116, 150, 227, 284, 408, 520, 718, 895, 1226, 1508, 2018, 2487, 3248, 3968, 5160, 6235, 7970, 9653, 12179, 14630, 18367, 21924, 27241, 32506, 39985, 47492, 58203, 68752, 83613, 98730, 119269, 140224, 168799, 197758, 236753, 277052, 329867, 384852, 457006, 531500, 628338
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..100
- C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, 2013.
- C. Carre, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, and O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015) 15.11.4.
Formula
The generating function for the numbers of espaliers of height h and volumes v_1 , ... v_h is x_1^{n_1} * ... x_h^{n_h} / ((1-x_1^{n_1}) *(1-x_1^{n_1}*x_2^{n_2}) *... *(1-x_1^{n_1}*x_2^{n_2}*...x_h^{n_h})).
This sequence is obtained with x_1 = ... = x_h = p by summing over n_1>= ... >= n_h>=1 and then over h.
Extensions
a(0)=1 prepended by Seiichi Manyama, Aug 20 2020
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