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%I #9 May 12 2019 02:23:28
%S 1,1,1,1,8,1,1,68,68,1,1,374,2340,374,1,1,1742,47012,47012,1742,1,1,
%T 7524,717948,2942288,717948,7524,1,1,31320,9259560,122248688,
%U 122248688,9259560,31320,1,1,127946,106900560,3895086794,12203119800,3895086794
%N Triangle T(n, m) = ( |-A008292(n+1,m+1)^2 + 2*binomial(n, m)^2| + A008292(n+1,m+1)*binomial(n, m) )/2 read by rows.
%C Row sums are 1, 2, 10, 138, 3090, 97510, 4393234, 263079138, 20207350402, 1926722077422, 223339810806978, ...
%F T(n, m) = T(n,n-m).
%e The triangle starts in row n=0 with columns 0 <= m <= n as
%e 1;
%e 1, 1;
%e 1, 8, 1;
%e 1, 68, 68, 1;
%e 1, 374, 2340, 374, 1;
%e 1, 1742, 47012, 47012, 1742, 1;
%e 1, 7524, 717948, 2942288, 717948, 7524, 1;
%e 1, 31320, 9259560, 122248688, 122248688, 9259560, 31320, 1;
%e 1, 127946, 106900560, 3895086794, 12203119800, 3895086794, 106900560, 127946, 1};
%p A178048 := proc(n,m) binomial(n,m)*A008292(n+1,m+1)+abs( -A008292(n+1,m+1)^2+2*binomial(n,m)^2) ; %/2; end proc:
%p seq(seq(A178048(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Nov 26 2010
%t << DiscreteMath`Combinatorica`
%t t[n_, m_] = (Abs[2*Binomial[n, m]^2 - Eulerian[n + 1, m]^2] + Binomial[n, m]*Eulerian[n + 1, m])/2;
%t Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t Flatten[%]
%Y Cf. A141686, A008292.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, May 18 2010
%E Definition corrected by _R. J. Mathar_, Nov 26 2010
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