A185962 Riordan array ((1-x)^2/(1-x+x^2), x(1-x)^2/(1-x+x^2)).

1, -1, 1, -1, -2, 1, 0, -1, -3, 1, 1, 2, 0, -4, 1, 1, 3, 5, 2, -5, 1, 0, 0, 3, 8, 5, -6, 1, -1, -4, -6, -1, 10, 9, -7, 1, -1, -4, -10, -16, -10, 10, 14, -8, 1, 0, 1, 0, -10, -26, -24, 7, 20, -9, 1, 1, 6, 15, 20, 5, -30, -42, 0, 27, -10, 1

Offset: 0

Author

Paul Barry, Feb 07 2011

Keywords

• sign
• easy
• tabl

Comments

Riordan array (g(x),xg(x)) where g(x)=(1-x)(1-x^2)(1-x^3)/(1-x^6).

Inverse is A185967. Row sums are A185963.

Diagonal sums are A185964. Central coefficients are A185965.

Subtriangle of the triangle given by (0, -1, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012

Examples

			Triangle begins:
   1;
  -1,  1;
  -1, -2,   1;
   0, -1,  -3,   1;
   1,  2,   0,  -4,   1;
   1,  3,   5,   2,  -5,   1;
   0,  0,   3,   8,   5,  -6,   1;
  -1, -4,  -6,  -1,  10,   9,  -7,  1;
  -1, -4, -10, -16, -10,  10,  14, -8,  1;
   0,  1,   0, -10, -26, -24,   7, 20, -9,   1;
   1,  6,  15,  20,   5, -30, -42,  0, 27, -10, 1;
  ...
From _Philippe Deléham_, Mar 19 2012: (Start)
(0, -1, 2, -1/2, 1/2, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0, -1,  1;
  0, -1, -2,  1;
  0,  0, -1, -3,  1;
  0,  1,  2,  0, -4,  1;
  0,  1,  3,  5,  2, -5,  1;
  ... (End)
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A195339, A195350.

Programs

Mathematica

CoefficientList[CoefficientList[Series[1/(1 - y*x + x/(1 - x)^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jul 23 2017 *)

Formula

T(n,k) = Sum_{i=0..(2*k+2)} C(2*k+2,i)*Sum_{j=0..(n-k-i)} C(k+j,j)*C(j,n-k-i-j)*(-1)^(n-k-j).

G.f.: 1/(1-y*x+x/(1-x)^2). - Philippe Deléham, Feb 07 2012

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = -1, T(2,1) = -2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham , Nov 11 2013