A131085 Triangle T(n,k) (n>=0, 0<=k<=n-1) read by rows, A007318 * A129686.

1, 1, 1, 0, 2, 1, -2, 2, 3, 1, -5, 0, 5, 4, 1, -9, -5, 5, 9, 5, 1, -14, -14, 0, 14, 14, 6, 1, -20, -28, -14, 14, 28, 20, 7, 1, -27, -48, -42, 0, 42, 48, 27, 8, 1, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1, -44, -110, -165, -132, 0, 132, 165, 110, 44, 10, 1

Offset: 0

Author

Gary W. Adamson, Jun 14 2007

Keywords

• sign
• tabl

Comments

Row sums = n.

A131085 * A000012 = A074909 starting (1, 2, 1, 3, 3, ...) instead of (1, 1, 2, 1, 3, 3, ...).

Examples

			First few rows of the triangle are:
   1;
   1,  1;
   0,  2, 1;
  -2,  2, 3, 1;
  -5,  0, 5, 4, 1;
  -9, -5, 5, 9, 5, 1;
-14, -14, 0, 14, 14, 6, 1;
-20, -28, -14, 14, 28, 20, 7, 1;
-27, -48, -42, 0, 42, 48, 27, 8, 1;
-35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
   ...
		

Crossrefs

Cf. A007318, A009766, A097808, A112466, A214292, A112467, A129686, A131084.

Programs

PARI

tabl(nn) = {t007318 = matrix(nn, nn, n, k, binomial(n-1, k-1)); t129686 = matrix(nn, nn, n, k, (k<=n)*((-1)^((n-k)\2)*((k==n) || (-1)*(k==(n-2))))); t131085 = t007318*t129686; for (n = 1, nn, for (k = 1, n, print1(t131085[n, k], ", ");););} \\ Michel Marcus, Feb 12 2014

Formula

Binomial transform of A129686 signed with (1, 1, 1, ...) in the main diagonal and (-1, -1, -1, ...) in the subsubdiagonal.

T(n,m) = T(n-1,m-1) + T(n-1,m). - Yuchun Ji, Dec 17 2018

T(2*k,k-1) = 0 for k > 0. - Yuchun Ji, Dec 20 2018

Comparing this triangle with the Catalan triangle A009766 one can see many similarities. For example, T(k+j,k) = A009766(k+1,j) for j < k+2. - Yuchun Ji, Dec 23 2018 [Edited by N. J. A. Sloane, Feb 11 2019]

Extensions

Missing comma corrected by Naruto Canada, Aug 26 2007

More terms from Michel Marcus, Feb 12 2014

Offset changed by N. J. A. Sloane, Feb 11 2019