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%I #8 Oct 05 2018 10:38:09
%S 0,0,1,0,2,1,2,1,4,3,5,4,7,6,8,7,12,11,15,14,20,19,24,23,31,30,37,36,
%T 45,44,52,51,64,63,75,74,90,89,104,103,124,123,143,142,167,166,190,
%U 189,221,220,251,250,288,287,324,323,369,368,413,412,465,464,516,515,580
%N G.f.: Sum_{n >= 1} ( x^(2^n)/ ((1+x^(2^(n-1)))*Product_{j=0..n-1} (1-x^(2^j)) ) ).
%C This is the g.f. for the number of non-squashing partitions with a repeated part, that is, A000123(n) - A088567(n).
%H N. J. A. Sloane and J. A. Sellers, <a href="http://arXiv.org/abs/math.CO/0312418">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.
%e x^2/((1 + x)*(1 - x)) + x^4/((1 + x^2)*(1 - x)*(1 - x^2)) + x^8/((1 + x^4)*(1 - x)*(1 - x^2)*(1 - x^4)) + ...
%t max = 65; Sum[x^(2^n)/((1+x^(2^(n-1))) Product[1-x^(2^j), {j, 0, n-1}]), {n, 1, Log[2, max] // Ceiling}] + O[x]^(max) // CoefficientList[#, x]& (* _Jean-François Alcover_, Oct 05 2018 *)
%Y Apart from initial terms, same as A088980.
%K nonn
%O 0,5
%A _N. J. A. Sloane_, Dec 02 2003
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