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A089054 Solution to the non-squashing boxes problem (version 1).

%I A089054 #17 Nov 04 2020 06:14:54
%S A089054 1,2,4,8,14,23,36,54,78,109,149,199,262,339,434,548,686,849,1043,1269,
%T A089054 1535,1842,2199,2607,3078,3613,4225,4915,5700,6581,7576,8686,9934,
%U A089054 11321,12871,14585,16493,18596,20925,23481,26303,29392,32788,36492,40553,44972,49799
%N A089054 Solution to the non-squashing boxes problem (version 1).
%C A089054 Given n boxes labeled 1..n, such that box i weighs i grams and can support a total weight of i grams; a(n) = number of stacks of boxes that can be formed such that no box is squashed.
%H A089054 Amanda Folsom, Youkow Homma, Jun Hwan Ryu, and Benjamin Tong, <a href="https://doi.org/10.1016/j.disc.2015.12.019">On a general class of non-squashing partitions</a>, Discrete Mathematics 339 (2016) 1482-1506.
%H A089054 Oystein J. Rodseth, <a href="https://doi.org/10.1016/j.disc.2006.03.051">Sloane's box stacking problem</a>, Discrete Math. 306 (2006), no. 16, 2005-2009.
%H A089054 Øystein J. Rødseth and James A. Sellers, <a href="https://ajc.maths.uq.edu.au/pdf/44/ajc_v44_p255.pdf">Congruences modulo high powers of 2 for Sloane's box stacking function</a>, Australasian Journal of Combinatorics, Volume 44 (2009), Pages 255-263.
%H A089054 N. J. A. Sloane and J. A. Sellers, <a href="https://arxiv.org/abs/math/0312418">On non-squashing partitions</a>, arXiv:math/0312418 [math.CO], 2003; Discrete Math., 294 (2005), 259-274.
%F A089054 G.f.: (B(x)-x)/(1-x)^2, where B(x) = g.f. for A088567.
%t A089054 max = 50; B[x_] = 1+x/(1-x) + Sum[x^(3 2^(k-1))/Product[(1-x^(2^j)), {j, 0, k}], {k, 1, Log[2, max]}] + O[x]^max;
%t A089054 A[x_] = (B[x]-x)/(1-x)^2;
%t A089054 CoefficientList[A[x], x] (* _Jean-François Alcover_, Sep 01 2018 *)
%Y A089054 Cf. A000123, A088567, A089055, A090631, A090632. Row sums of A090641.
%K A089054 nonn
%O A089054 0,2
%A A089054 _N. J. A. Sloane_, Dec 04 2003