A292781 Triangle read by rows: T(n,k) = T(n-k,k-1) with T(0,0) = 1 and T(n,0) = -1/n * Sum_{k=1..A003056(n)} (-1)^k * (2*k+1) * (n+1-A060544(k+1)) * T(n,k).

1, -24, 1, 252, -24, -1472, 252, 1, 4830, -1472, -24, -6048, 4830, 252, -16744, -6048, -1472, 1, 84480, -16744, 4830, -24, -113643, 84480, -6048, 252, -115920, -113643, -16744, -1472, 534612, -115920, 84480, 4830, 1, -370944, 534612, -113643, -6048, -24

Offset: 0

Author

Seiichi Manyama, Sep 23 2017

Keywords

• sign
• tabf

Examples

			First few rows are:
        1;
      -24,       1;
      252,     -24;
    -1472,     252,      1;
     4830,   -1472,    -24;
    -6048,    4830,    252;
   -16744,   -6048,  -1472,     1;
    84480,  -16744,   4830,   -24;
  -113643,   84480,  -6048,   252;
  -115920, -113643, -16744, -1472;
   534612, -115920,  84480,  4830, 1.
-----------------------------------------
n=5
T(5,1) = T(4,0) = 4830,  T(5,2) = T(3,1) = 252.
T(5,0) = -1/5 * Sum_{k=1..2} (-1)^k * (2*k+1) * (5+1-A060544(k+1)) * T(n,k) = -1/5 * ((-3)*(-4)*4830 + 5*(-22)*252) = -6048.
n=6
T(6,1) = T(5,0) = -6048,  T(6,2) = T(4,1) = -1472, T(6,3) = T(3,2) = 1.
T(6,0) = -1/6 * Sum_{k=1..3} (-1)^k * (2*k+1) * (6+1-A060544(k+1)) * T(n,k) = -1/6 * ((-3)*(-3)*(-6048) + 5*(-21)*(-1472) - 7*(-48)*1) = -16744.
		

Links

• Seiichi Manyama, Rows n = 0..481, flattened

Crossrefs

All columns are A000594.

Cf. A003056, A060544.

Programs

Ruby

def A292781(n)
  ary = [[1]]
  (1..n).each{|i|
    m = ((Math.sqrt(1 + 8 * i) - 1) / 2).to_i
    a = (1..m).map{|j| ary[i - j][j - 1]}
    ary << [-(1..m).inject(0){|s, j| s + (-1) ** (j % 2) * (2 * j + 1) * (i - 9 * j * (j + 1) / 2) * a[j - 1]} / i] + a
  }
  ary.flatten
end
p A292781(20)