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Riordan array (1/(1+2xc(-2x)),xc(-2x)/(1+2xc(-2x))), c(x) the g.f. of A000108.
4

%I #17 Dec 29 2023 13:10:05

%S 1,-2,1,8,-6,1,-40,36,-10,1,224,-224,80,-14,1,-1344,1440,-600,140,-18,

%T 1,8448,-9504,4400,-1232,216,-22,1,-54912,64064,-32032,10192,-2184,

%U 308,-26,1,366080,-439296,232960,-81536,20160,-3520,416,-30,1,-2489344,3055104,-1697280,639744,-176256,35904,-5304,540,-34,1

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

%C Row sums are A114191. Diagonal sums are A114194. Inverse of A114192.

%C 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 Riordan array ((sqrt(1+8x)-1)/(4x), (sqrt(1+8x)-1)^2/(16x)).

%F 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 Triangle begins

%e 1;

%e -2, 1;

%e 8, -6, 1;

%e -40, 36, -10, 1;

%e 224, -224, 80, -14, 1;

%e -1344, 1440, -600, 140, -18, 1;

%t c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);

%t (* The function RiordanArray is defined in A256893. *)

%t RiordanArray[1/(1 + 2 # c[-2 #])&, # c[-2 #]/(1 + 2 # c[-2 #])&, 10] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)

%Y Cf. A039599, A084938.

%K easy,sign,tabl

%O 0,2

%A _Paul Barry_, Nov 16 2005