A380191 Triangle read by rows: Riordan array (2 - D(x), x * D(x)) where D(x) is g.f. of A001764.

1, -1, 1, -3, 0, 1, -12, -1, 1, 1, -55, -6, 2, 2, 1, -273, -33, 5, 6, 3, 1, -1428, -182, 13, 22, 11, 4, 1, -7752, -1020, 28, 91, 46, 17, 5, 1, -43263, -5814, 0, 408, 210, 78, 24, 6, 1, -246675, -33649, -627, 1938, 1020, 380, 119, 32, 7, 1, -1430715, -197340, -6325, 9614, 5187, 1938, 612, 170, 41, 8, 1

Offset: 0

Author

Werner Schulte, Jan 15 2025

Keywords

• sign
• easy
• tabl

Examples

			Triangle T(n, k) for 0 <= k <= n starts:
n \k :         0        1      2     3     4     5    6    7   8  9  10
=======================================================================
   0 :         1
   1 :        -1        1
   2 :        -3        0      1
   3 :       -12       -1      1     1
   4 :       -55       -6      2     2     1
   5 :      -273      -33      5     6     3     1
   6 :     -1428     -182     13    22    11     4    1
   7 :     -7752    -1020     28    91    46    17    5    1
   8 :    -43263    -5814      0   408   210    78   24    6   1
   9 :   -246675   -33649   -627  1938  1020   380  119   32   7  1
  10 :  -1430715  -197340  -6325  9614  5187  1938  612  170  41  8   1
  etc.
		

Crossrefs

Cf. A001764, A110616.

Programs

PARI

T(n, k) = if(k==n, 1, binomial(3*n-2*k, n-k) * (n*k+4*k-3*n) / ((3*n-2*k) * (2*n-k+1)))

Formula

T(n, k) = binomial(3*n - 2*k, n - k) * (n*k + 4*k - 3*n) / ((3*n - 2*k) * (2*n - k + 1)) if 0 <= k < n, and T(n, n) = 1 for n >= 0.

G.f.: (2 - D(t)) / (1 - x * t * D(t)) where D(t) is g.f. of A001764.

Conjecture: Sum_{i=0..n-k} binomial(2*i, i) * T(n, i+k) = A110616(n, k).