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A097808 Riordan array ((1+2x)/(1+x)^2, 1/(1+x)) read by rows.

%I A097808 #15 Jul 16 2019 10:38:28
%S A097808 1,0,1,-1,-1,1,2,0,-2,1,-3,2,2,-3,1,4,-5,0,5,-4,1,-5,9,-5,-5,9,-5,1,6,
%T A097808 -14,14,0,-14,14,-6,1,-7,20,-28,14,14,-28,20,-7,1,8,-27,48,-42,0,42,
%U A097808 -48,27,-8,1,-9,35,-75,90,-42,-42,90,-75,35,-9,1,10,-44,110,-165,132,0,-132,165,-110,44,-10,1
%N A097808 Riordan array ((1+2x)/(1+x)^2, 1/(1+x)) read by rows.
%C A097808 Inverse of A059260. Row sums are inverse binomial transform of A040000, with g.f. (1+2x)/(1+x). Diagonal sums are (-1)^n(1-Fib(n)). A097808=B^(-1)*A097806, where B is the binomial matrix. B*A097808*B^(-1) is the inverse of A097805.
%H A097808 Robert Israel, <a href="/A097808/b097808.txt">Table of n, a(n) for n = 0..10010</a>  (rows 0 to 140, flattened)
%F A097808 Columns have g.f. (1+2x)/(1+x)^2(x/(1+x))^k.
%F A097808 T(n,k)=T(n-1,k-1)-3*T(n-1,k)+2*T(n-2,k-1)-3*T(n-2,k)+T(n-3,k-1)-T(n-3,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=0, T(2,0)=T(2,1)=-1, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham_, Jan 12 2014
%F A097808 T(0,0)=1, T(n,0)=(-1)^(n-1)*(n-1) for n>0, T(n,n)=1, T(n,k)=T(n-1,k-1)-T(n-1,k) for 0<k<n. - _Philippe Deléham_, Jan 12 2014
%e A097808 Rows begin
%e A097808 1;
%e A097808 0, 1;
%e A097808 -1, -1, 1;
%e A097808 2, 0, -2, 1;
%e A097808 -3, 2, 2, -3, 1;
%e A097808 4, -5, 0, 5, -4, 1;
%e A097808 -5, 9, -5, -5, 9, -5, 1;
%e A097808 6, -14, 14, 0, -14, 14, -6, 1;
%e A097808 -7, 20, -28, 14, 14, -28, 20, -7, 1;
%e A097808 8, -27, 48, -42, 0, 42, -48, 27, -8, 1;
%p A097808 T:= proc(n,k) option remember;
%p A097808 if k < 0 or k > n then return 0 fi;
%p A097808 procname (n-1,k-1)-3*procname(n-1,k)+2*procname(n-2,k-1)-3*procname(n-2,k)+
%p A097808 procname(n-3,k-1)-procname(n-3,k)
%p A097808 end proc:
%p A097808 T(0,0):= 1: T(1,1):= 1: T(2,2):= 1:
%p A097808 T(1,0):= 0: T(2,0):= -1: T(2,1):= -1:
%p A097808 seq(seq(T(n,k),k=0..n),n=0..12); # _Robert Israel_, Jul 16 2019
%t A097808 (* The function RiordanArray is defined in A256893. *)
%t A097808 RiordanArray[(1 + 2 #)/(1 + #)^2&, #/(1 + #)&, 12] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)
%K A097808 easy,sign,tabl
%O A097808 0,7
%A A097808 _Paul Barry_, Aug 25 2004