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%I #6 Jan 21 2017 10:48:04
%S 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,2,6,2,1,1,1,1,2,6,6,2,
%T 1,1,1,1,2,6,24,6,2,1,1,1,1,2,6,24,24,6,2,1,1,1,1,2,6,24,120,24,6,2,1,
%U 1,1,1,2,6,24,120,120,24,6,2,1,1,1,1,2,6,24,120,720,120,24,6,2,1,1,1,1,2,6,24
%N Triangle T(n,m) read by rows: m! if m <= floor(n/2), and (n-m)! otherwise.
%C Row sums: 1, 2, 3, 4, 6, 8, 14, 20, 44, 68, 188,... which is
%C 2*A003422((n+1)/2) if n is odd, and A003422(n/2)+A003422(1+n/2) if n is even.
%F T(n,m) = A000142(m) if m<=[n/2], = A000142(n-m) if m>[n/2], 0<=m<=n.
%F Conjecture: limit_{n->infinity} sum_{m=0..n} ( 1/binomial(n,m) - T(n,m) ) = 0.
%F T(n,m) = T(n,n-m).
%e 1,
%e 1, 1,
%e 1, 1, 1,
%e 1, 1, 1, 1,
%e 1, 1, 2, 1, 1,
%e 1, 1, 2, 2, 1, 1,
%e 1, 1, 2, 6, 2, 1, 1,
%e 1, 1, 2, 6, 6, 2, 1, 1,
%e 1, 1, 2, 6, 24, 6, 2, 1,1,
%e 1, 1, 2, 6, 24, 24, 6, 2, 1, 1,
%e 1, 1, 2, 6, 24, 120, 24, 6, 2, 1, 1
%t g[n_, m_] := If[m <= Floor[n/2], m!, (n - m)! ]; w = Table[Table[g[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[w] Table[Apply[Plus, Table[g[n, m], {m, 0, n}]], {n, 0, 10}]
%K nonn,easy,tabl
%O 0,13
%A _Roger L. Bagula_ and _Gary W. Adamson_, May 30 2008
%E Non-Ascii characters corrected, offset set to 0, reported Mma experiments removed - The Assoc. Editors of the OEIS, Oct 31 2009
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