A177428
Triangle T(n,m)= A141686(n,m)*(m-1)! read by rows, n>=1, 1<=m<=n.
Original entry on oeis.org
1, 1, 1, 1, 8, 2, 1, 33, 66, 6, 1, 104, 792, 624, 24, 1, 285, 6040, 18120, 6840, 120, 1, 720, 35730, 289920, 428760, 86400, 720, 1, 1729, 180306, 3279990, 13119960, 10818360, 1244880, 5040, 1, 4016, 818048, 29646624, 262399200, 592932480, 294497280
Offset: 1
1;
1, 1;
1, 8, 2;
1, 33, 66, 6;
1, 104, 792, 624, 24;
1, 285, 6040, 18120, 6840, 120;
1, 720, 35730, 289920, 428760, 86400, 720;
1, 1729, 180306, 3279990, 13119960, 10818360, 1244880, 5040;
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A177428 := proc(n,k)
A008292(n,k)*(n-1)!/(n-k)! ;
end proc: # R. J. Mathar, May 15 2016
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<< DiscreteMath`Combinatorica`
t[n_, m_] = Eulerian[n + 1, m]*n!/(n - m)!;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]
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.
Original entry on oeis.org
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
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;
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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
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<< 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[%]
A178046
Triangle t(n, m) = 2*binomial(n,m)^2 -A008292(n+1,m+1)^2 read by rows.
Original entry on oeis.org
1, 1, 1, 1, -8, 1, 1, -103, -103, 1, 1, -644, -4284, -644, 1, 1, -3199, -91004, -91004, -3199, 1, 1, -14328, -1418031, -5836256, -1418031, -14328, 1, 1, -60911, -18428967, -243950711, -243950711, -18428967, -60911, 1, 1, -251876
Offset: 0
1;
1, 1;
1, -8, 1;
1, -103, -103, 1;
1, -644, -4284, -644, 1;
1, -3199, -91004, -91004, -3199, 1;
1, -14328, -1418031, -5836256, -1418031, -14328, 1;
1, -60911, -18428967, -243950711, -243950711, -18428967, -60911, 1;
1, -251876, -213392096, -7785232484, -24395306300, -7785232484, -213392096, -251876, 1;
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<< DiscreteMath`Combinatorica`
t[n_, m_] = 2*Binomial[n, m]^2 - Eulerian[n + 1, m]^2;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]
Showing 1-3 of 3 results.
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