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A114193 Riordan array (1/(1+2xc(-2x)),xc(-2x)/(1+2xc(-2x))), c(x) the g.f. of A000108.

%I A114193 #17 Dec 29 2023 13:10:05
%S A114193 1,-2,1,8,-6,1,-40,36,-10,1,224,-224,80,-14,1,-1344,1440,-600,140,-18,
%T A114193 1,8448,-9504,4400,-1232,216,-22,1,-54912,64064,-32032,10192,-2184,
%U A114193 308,-26,1,366080,-439296,232960,-81536,20160,-3520,416,-30,1,-2489344,3055104,-1697280,639744,-176256,35904,-5304,540,-34,1
%N A114193 Riordan array (1/(1+2xc(-2x)),xc(-2x)/(1+2xc(-2x))), c(x) the g.f. of A000108.
%C A114193 Row sums are A114191. Diagonal sums are A114194. Inverse of A114192.
%C A114193 Triangle T(n,k), read by rows, given by (-2, -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_, Jan 26 2014
%F A114193 Riordan array ((sqrt(1+8x)-1)/(4x), (sqrt(1+8x)-1)^2/(16x)).
%F A114193 T(n, k) = (-2)^(n-k)*A039599(n, k) = (-2)^(n-k)*C(2*n, n-k)*(2*k+1)/(n+k+1). - _Philippe Deléham_, Nov 17 2005
%e A114193 Triangle begins
%e A114193       1;
%e A114193      -2,    1;
%e A114193       8,   -6,    1;
%e A114193     -40,   36,  -10,   1;
%e A114193     224, -224,   80, -14,   1;
%e A114193   -1344, 1440, -600, 140, -18, 1;
%t A114193 c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);
%t A114193 (* The function RiordanArray is defined in A256893. *)
%t A114193 RiordanArray[1/(1 + 2 # c[-2 #])&, # c[-2 #]/(1 + 2 # c[-2 #])&, 10] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)
%Y A114193 Cf. A039599, A084938.
%K A114193 easy,sign,tabl
%O A114193 0,2
%A A114193 _Paul Barry_, Nov 16 2005