A113313 - OEIS

%I #13 Apr 17 2022 22:51:15

%S 1,-2,1,0,-1,1,0,-1,0,1,0,-1,-1,1,1,0,-1,-2,0,2,1,0,-1,-3,-2,2,3,1,0,

%T -1,-4,-5,0,5,4,1,0,-1,-5,-9,-5,5,9,5,1,0,-1,-6,-14,-14,0,14,14,6,1,0,

%U -1,-7,-20,-28,-14,14,28,20,7,1,0,-1,-8,-27,-48,-42,0,42,48,27,8,1,0,-1,-9,-35

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

%C Row sums are (1,-1,0,0,0,...) = 2*C(0,n) - C(1,n).

%C Diagonal sums are -2*0^n - F(n-4) with g.f. (1 - 3x + 2x^2) / (1 - x - x^2).

%C Inverse of A113310.

%F T(n, k) = C(n-1, n-k) - 2*C(n-2, n-k-1).

%F exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-x + x^3/3!) = -x - 2*x^2/2! - 2*x^3/3! + 5*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - _Peter Bala_, Dec 21 2014

%e The triangle T(n, k) begins:

%e n\k 0  1  2   3   4   5  6  7  8 9 10 ...

%e 0:  1

%e 1: -2  1

%e 2:  0 -1  1

%e 3:  0 -1  0   1

%e 4:  0 -1 -1   1   1

%e 5:  0 -1 -2   0   2   1

%e 6:  0 -1 -3  -2   2   3  1

%e 7:  0 -1 -4  -5   0   5  4  1

%e 8:  0 -1 -5  -9  -5   5  9  5  1

%e 9:  0 -1 -6 -14 -14   0 14 14  6 1

%e 10: 0 -1 -7 -20 -28 -14 14 28 20 7  1

%e ... Reformatted. - _Wolfdieter Lang_, Jan 06 2015

%Y Cf. A113310.

%K easy,sign,tabl

%O 0,2

%A _Paul Barry_, Oct 25 2005