Geodesic
On the distribution of the number of ones in a Boolean Pascal's triangle
Diskretnaya Matematika, Tome 18 (2006) no. 2, pp. 123-131.

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This research is devoted to estimating the number of Boolean Pascal's triangles of large enough size $s$ containing a given number of ones $\xi\le ks$, $k>0$. We demonstrate that any such Pascal's triangle contains a zero triangle whose size differs from $s$ by at most constant depending only on $k$. We prove that there is a monotone unbounded sequence of rational numbers $0=k_0$ such that the distribution of the number of triangles is concentrated in some neighbourhoods of the points $k_is$. The form of the distribution in each neighbourhood depends not on $s$ but on the residue of $s$ some modulo depending on $i\ge 0$. 
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F. M. Malyshev; E. V. Kutyreva. On the distribution of the number of ones in a Boolean Pascal's triangle. Diskretnaya Matematika, Tome 18 (2006) no. 2, pp. 123-131. http://geodesic.mathdoc.fr/item/DM_2006_18_2_a8/

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