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

0,12

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.

Links

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

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

Example

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;

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 *)

Crossrefs

Cf. A206474 (unsigned version).

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

Sequence in context: A165031 A286634 A099245 * A206474 A211999 A175025

Adjacent sequences: A185328 A185329 A185330 * A185332 A185333 A185334

Keyword

easy,sign,tabl

Author

Philippe Deléham, Feb 08 2012

Status

approved