A178048 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.

1, 1, 1, 1, 8, 1, 1, 68, 68, 1, 1, 374, 2340, 374, 1, 1, 1742, 47012, 47012, 1742, 1, 1, 7524, 717948, 2942288, 717948, 7524, 1, 1, 31320, 9259560, 122248688, 122248688, 9259560, 31320, 1, 1, 127946, 106900560, 3895086794, 12203119800, 3895086794, 106900560, 127946, 1

Offset: 0

Author

Roger L. Bagula, May 18 2010

Keywords

• nonn
• tabl

Examples

			The triangle starts in row n=0 with columns 0 <= m <= n as
  1;
  1,      1;
  1,      8,         1;
  1,     68,        68,          1;
  1,    374,      2340,        374,           1;
  1,   1742,     47012,      47012,        1742,          1;
  1,   7524,    717948,    2942288,      717948,       7524,         1;
  1,  31320,   9259560,  122248688,   122248688,    9259560,     31320,      1;
  1, 127946, 106900560, 3895086794, 12203119800, 3895086794, 106900560, 127946, 1;
		

Crossrefs

Cf. A141686, A008292.

Programs

Maple

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:
seq(seq(A178048(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Nov 26 2010

Mathematica

<< DiscreteMath`Combinatorica`
t[n_, m_] = (Abs[2*Binomial[n, m]^2 - Eulerian[n + 1, m]^2] + Binomial[n, m]*Eulerian[n + 1, m])/2;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

T(n, m) = T(n,n-m).

Extensions

Definition corrected by R. J. Mathar, Nov 26 2010