A185331 Riordan array ((1-x+x^2)/(1+x^2), x/(1+x^2)).

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 0, 2, -2, -1, 1, -1, 1, 3, -3, -1, 1, 0, -3, 3, 4, -4, -1, 1, 1, -1, -6, 6, 5, -5, -1, 1, 0, 4, -4, -10, 10, 6, -6, -1, 1, -1, 1, 10, -10, -15, 15, 7, -7, -1, 1, 0, -5, 5, 20, -20, -21, 21, 8, -8, -1, 1

Offset: 0

Author

Philippe Deléham, Feb 08 2012

Keywords

• easy
• sign
• tabl

Comments

Triangle T(n,k), read by rows, given by (-1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Examples

			Triangle begins:
   1;
  -1,  1;
   0, -1,   1;
   1, -1,  -1,   1;
   0,  2,  -2,  -1,   1;
  -1,  1,   3,  -3,  -1,   1;
   0, -3,   3,   4,  -4,  -1,   1;
   1, -1,  -6,   6,   5,  -5,  -1,  1;
   0,  4,  -4, -10,  10,   6,  -6, -1,  1;
  -1,  1,  10, -10, -15,  15,   7, -7, -1,  1;
   0, -5,   5,  20, -20, -21,  21,  8, -8, -1,  1;
   1, -1, -15,  15,  35, -35, -28, 28,  9, -9, -1, 1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Crossrefs

Cf. A206474 (unsigned version).

Cf. A007318, A053122, A078812, A085478, A128908, A129818,

Programs

Mathematica

CoefficientList[Series[CoefficientList[Series[(1 - x + x^2)/(1 - y*x + x^2), {x, 0, 10}], x], {y, 0, 10}], y] // Flatten (* G. C. Greubel, Jun 27 2017 *)

Formula

T(n,k) = T(n-1,k-1) - T(n-2,k), T(0,0) = 1, T(0,1) = -1, T(0,2) = 0.

G.f.: (1-x+x^2)/(1-y*x+x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A184334(n), A163805(n), A000007(n), A028310(n), A025169(n-1), A005320(n) (n>0) for x = -1, 0, 1, 2, 3, 4 respectively.

T(n,n) = 1, T(n+1,n) = -1, T(n+2,n) = -n, T(n+3,n) = n+1, T(n+4,n) = n(n+1)/2 = A000217(n).

T(2n,2k) = (-1)^(n-k) * A128908(n,k), T(2n+1,2k+1) = -T(2n+1,2k) = A129818(n,k), T(2n+2,2k+1) = (-1)*A053122(n,k). - Philippe Deléham, Feb 09 2012