A099096 Riordan array (1,2-x).

1, 0, 2, 0, -1, 4, 0, 0, -4, 8, 0, 0, 1, -12, 16, 0, 0, 0, 6, -32, 32, 0, 0, 0, -1, 24, -80, 64, 0, 0, 0, 0, -8, 80, -192, 128, 0, 0, 0, 0, 1, -40, 240, -448, 256, 0, 0, 0, 0, 0, 10, -160, 672, -1024, 512, 0, 0, 0, 0, 0, -1, 60, -560, 1792, -2304, 1024, 0, 0, 0, 0, 0

Offset

0,3

Comments

Row sums are n+1 = Sum_{k=0..n} binomial(k,n-k)*2^(2k-n)*(-1)^(n-k). Diagonal sums are (-1)^n*A008346(n). The Riordan array (1,s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, -1/2, 1/2, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008

Links

Table of n, a(n) for n=0..70.

Formula

Number triangle T(n, k) = binomial(k, n-k)*2^k*(-1/2)^(n-k); columns have g.f. (2x-x^2)^k.

G.f. of column k of matrix power T^m = (1 - (1-x)^(2^m))^k for k >= 0, when including the leading zeros that appear above the diagonal. - Paul D. Hanna, Nov 15 2007

T(n,k) = 2*T(n-1,k-1) - T(n-2,k-1), with T(0,0)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 25 2013

G.f.: 1/(1-2*x*y+x^2*y). - R. J. Mathar, Aug 12 2015

Example

Rows begin
  1;
  0,   2;
  0,  -1,   4;
  0,   0,  -4,   8;
  0,   0,   1, -12,  16;
  ...

Prog

(PARI) /* Matrix power T^m formula: [T^m](n, k) = */ {T(n, k, m=1)=polcoeff((1 - (1-x +x*O(x^n))^(2^m) )^k, n)} \\ Paul D. Hanna, Nov 15 2007

Crossrefs

Cf. A099089.

Sequence in context: A271584 A072737 A061290 * A099089 A121298 A212206

Adjacent sequences: A099093 A099094 A099095 * A099097 A099098 A099099

Keyword

sign,easy,tabl

Author

Paul Barry, Sep 25 2004

Status

approved