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A229915 Number of espalier polycubes of a given volume in dimension 3. 5


%S 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 coordinate (0,b,c) belonging to the first plateau is 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 coordinate (a,0,0).

%D C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, 2013; http://hal.archives-ouvertes.fr/docs/00/90/58/89/PDF/polycubes.pdf

%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. A227925, A230118, A229917, A229925.

%K nonn

%O 1,2

%A _Matthieu Deneufchâtel_, Oct 03 2013

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Last modified March 23 16:50 EDT 2018. Contains 301123 sequences. (Running on oeis4.)