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%I #3 Oct 13 2012 14:48:20
%S 1,1,1,1,6,1,1,9,9,1,1,12,30,12,1,1,15,50,50,15,1,1,12,60,140,60,12,1,
%T 1,14,63,245,245,63,14,1,1,8,84,336,630,336,84,8,1,1,9,72,336,1008,
%U 1008,336,72,9,1,1,10,45,360,1470,2772,1470,360,45,10,1
%N A weighted binomial triangle with smooth weighting function: f(n,d) = Floor[1 + d*Sech[d/2 - n]].
%C Row sums are:
%C {1, 2, 4, 13, 46, 118, 275, 633, 1481, 2844, 6535}.;
%C Here the interior coefficients are larger than the
%C pascal triangle: most generalized Pascal triangles yield smaller interior
%C coefficients.
%F a(n,d)=If[n == 0 || n == d, 1, f[n, d]* Binomial[d, n]]; f(n,d) = Floor[1 + d*Sech[d/2 - n]].
%e {{1},
%e {1, 1},
%e {1, 6, 1},
%e {1, 9, 9, 1},
%e {1, 12, 30, 12, 1},
%e {1, 15, 50, 50, 15,1},
%e {1, 12, 60, 140, 60, 12, 1},
%e {1, 14, 63, 245, 245, 63, 14, 1},
%e {1, 8, 84, 336, 630, 336, 84, 8, 1},
%e {1, 9, 72, 336, 1008, 1008, 336, 72, 9,1},
%e {1, 10, 45, 360, 1470, 2772, 1470, 360, 45, 10, 1}}
%t f[n_, d_] = Floor[1 + d*Sech[d/2 - n]]; a = Table[Table[If[n == 0 || n == d, 1, f[n, d]* Binomial[ d, n]], {n, 0, d}], {d, 0, 10}]; Flatten[a] Table[Apply[Plus, Table[If[n == 1 || n == d, 1, f[n, d]* Binomial[d, n]], {n, 0, d}]], {d, 0, 10}];
%K nonn,uned,tabl
%O 1,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, May 23 2008
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