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A114189 Riordan array (1/(1+xc(-2x)), xc(-2x)/(1+xc(-2x))), c(x) the g.f. of A000108.

%I A114189 #16 Dec 29 2023 13:10:45
%S A114189 1,-1,1,3,-4,1,-13,19,-7,1,67,-102,44,-10,1,-381,593,-278,78,-13,1,
%T A114189 2307,-3640,1795,-568,121,-16,1,-14589,23231,-11849,4051,-999,173,-19,
%U A114189 1,95235,-152650,79750,-28770,7820,-1598,234,-22,1,-636925,1025965,-545680,204760,-59650,13642,-2392,304,-25,1
%N A114189 Riordan array (1/(1+xc(-2x)), xc(-2x)/(1+xc(-2x))), c(x) the g.f. of A000108.
%C A114189 Inverse of A114188. Factors as (1,xc(-2x))*(1/(1+x), x/(1+x)). Row sums are 0^n. Diagonal sums are A114190. First column is A114191. A signed version of A110506.
%C A114189 Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -1,-2,-2,-2,-2,-2,-2,...] DELTA [1,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 01 2007
%F A114189 Riordan array ((3-sqrt(1+8x))/(2(1-x)), (sqrt(1+8x)-2x-1)/(2(1-x))).
%F A114189 T(n,k) = (-1)^(n-k)*A110506(n,k). - _Philippe Deléham_, Mar 24 2007
%e A114189 Triangle begins
%e A114189      1;
%e A114189     -1,    1;
%e A114189      3,   -4,    1;
%e A114189    -13,   19,   -7,   1;
%e A114189     67, -102,   44, -10,   1;
%e A114189   -381,  593, -278,  78, -13, 1;
%t A114189 c[x_] := (1 - Sqrt[1 - 4x])/(2x);
%t A114189 (* The function RiordanArray is defined in A256893. *)
%t A114189 RiordanArray[1/(1 + # c[-2#])&, # c[-2#]/(1 + # c[-2#])&, 10] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)
%K A114189 easy,sign,tabl
%O A114189 0,4
%A A114189 _Paul Barry_, Nov 16 2005