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%I #7 Oct 10 2018 06:16:00
%S 1,1,2,4,14,26,107,190,1234,2182,9947,17414,126953,228398,1039404,
%T 1857419,19047146,35215110,168364007,307674658,2378963269,4429446046,
%U 20237375204,37371654467,410117798653,776233491226,3797821367602
%N Number of non-squashing partitions of {1,...,n}.
%C A "non-squashing" partition of n is one where n=p_1+p_2+...+p_k with 1 <= p_1 <= p_2 <= ... <= p_k and p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k.
%H N. J. A. Sloane and J. A. Sellers, <a href="http://arxiv.org/abs/math/0312418">On non-squashing partitions</a>, arXiv:math/0312418 [math.CO], 2003.
%H N. J. A. Sloane and J. A. Sellers, <a href="http://dx.doi.org/10.1016/j.disc.2004.11.014">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.
%F a(n) = Sum_{i = 0..ceiling(n/2)-1} (binomial(n, i)*a(i)) + [if n is even] binomial(n, n/2)*(a(n/2)-1/2).
%t a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n, i]*a[i], {i, 0, Ceiling[ n/2] - 1}] + If[EvenQ[n], Binomial[n, n/2] (a[n/2] - 1/2), 0];
%t Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Oct 10 2018 *)
%Y Cf. A018819, A115625.
%K nonn
%O 0,3
%A _Christian G. Bower_, Jan 26 2006