A099096 - OEIS

%I #20 Jan 25 2020 01:42:12

%S 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,

%T 0,0,0,0,-8,80,-192,128,0,0,0,0,1,-40,240,-448,256,0,0,0,0,0,10,-160,

%U 672,-1024,512,0,0,0,0,0,-1,60,-560,1792,-2304,1024,0,0,0,0,0

%N Riordan array (1,2-x).

%C 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).

%C 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

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

%F 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

%F 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

%F G.f.: 1/(1-2*x*y+x^2*y). - _R. J. Mathar_, Aug 12 2015

%e Rows begin

%e   1;

%e   0,   2;

%e   0,  -1,   4;

%e   0,   0,  -4,   8;

%e   0,   0,   1, -12,  16;

%e   ...

%o (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

%Y Cf. A099089.

%K sign,easy,tabl

%O 0,3

%A _Paul Barry_, Sep 25 2004