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A099096 Riordan array (1,2-x). 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified March 26 18:36 EDT 2019. Contains 321511 sequences. (Running on oeis4.)