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

Offset

0,5

Comments

Row sums are 1, 2, 10, 138, 3090, 97510, 4393234, 263079138, 20207350402, 1926722077422, 223339810806978, ...

Links

Table of n, a(n) for n=0..41.

Formula

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

Example

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};

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[%]

Crossrefs

Cf. A141686, A008292.

Sequence in context: A203443 A176642 A172346 * A174728 A015121 A156766

Adjacent sequences: A178045 A178046 A178047 * A178049 A178050 A178051

Keyword

nonn,tabl

Author

Roger L. Bagula, May 18 2010

Extensions

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

Status

approved