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

0,5

Comments

Row sums = n.

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

Links

Table of n, a(n) for n=0..65.

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]

Example

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;
   ...

Prog

(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

Crossrefs

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

Sequence in context: A336470 A351093 A070104 * A303280 A318717 A331887

Adjacent sequences: A131082 A131083 A131084 * A131086 A131087 A131088

Keyword

sign,tabl

Author

Gary W. Adamson, Jun 14 2007

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

Status

approved