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%I #52 Aug 22 2020 11:08:18
%S 1,1,3,5,10,14,26,34,57,76,116,150,227,284,408,520,718,895,1226,1508,
%T 2018,2487,3248,3968,5160,6235,7970,9653,12179,14630,18367,21924,
%U 27241,32506,39985,47492,58203,68752,83613,98730,119269,140224,168799,197758,236753,277052,329867,384852,457006,531500,628338
%N Number of espalier polycubes of a given volume in dimension 3.
%C A pyramid polycube is obtained by gluing together horizontal plateaux (parallelepipeds of height 1) in such a way that (0,0,0) belongs to the first plateau and each cell with coordinates (0,b,c) belongs to the first plateau such that b,c >= 0. If the cell with coordinates (a,b,c) belongs to the (a+1)-st plateau (a>0), then the cell with coordinates (a-1, b, c) belongs to the a-th plateau.
%C An espalier polycube is a special pyramid such that each plateau contains the cell with coordinates (a,0,0).
%H Seiichi Manyama, <a href="/A229915/b229915.txt">Table of n, a(n) for n = 0..100</a>
%H C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., <a href="https://hal.archives-ouvertes.fr/hal-00905889">Dirichlet convolution and enumeration of pyramid polycubes</a>, 2013.
%H C. Carre, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, and O. Mallet, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Dubernard/dub4.html">Enumeration of Polycubes and Dirichlet Convolutions</a>, J. Int. Seq. 18 (2015) 15.11.4.
%F 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})).
%F This sequence is obtained with x_1 = ... = x_h = p by summing over n_1>= ... >= n_h>=1 and then over h.
%Y Cf. A026820, A100882, A227925, A230118, A229917, A229925, A323582.
%K nonn
%O 0,3
%A _Matthieu Deneufchâtel_, Oct 03 2013
%E a(0)=1 prepended by _Seiichi Manyama_, Aug 20 2020
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