%I #16 Dec 29 2023 12:04:24

%S 1,-2,1,6,-5,1,-24,24,-8,1,114,-123,51,-11,1,-600,672,-312,87,-14,1,

%T 3372,-3858,1914,-618,132,-17,1,-19824,22992,-11904,4218,-1068,186,

%U -20,1,120426,-140991,75183,-28383,8043,-1689,249,-23,1,-749976,884112,-481704,190347,-58398,13929,-2508,321,-26,1

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

%C First column is (-1)^n*A054872(n). Row sums are a signed version of A108524. Inverse of generalized Delannoy triangle A121574. Unsigned triangle is A121576.

%C Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -2, -1, -3, -1, -3, -1, -3, -1, -3, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Aug 09 2006

%H G. C. Greubel, <a href="/A121575/b121575.txt">Rows n=0..100 of triangle, flattened</a>

%F T(n,k) = (-1)^(n-k)*(1/2)*Sum_{i=0..n-k} binomial(n,i) * binomial(2*n-k-i,n)*(4 - 9*i + 3*i^2 - 6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i. - _G. C. Greubel_, Nov 02 2018

%e Triangle begins

%e 1;

%e -2, 1;

%e 6, -5, 1;

%e -24, 24, -8, 1;

%e 114, -123, 51, -11, 1;

%e -600, 672, -312, 87, -14, 1;

%t Flatten[Table[(-1)^(n-k)*Sum[Binomial[n, i] Binomial[2*n-k-i, n]*(4-9*i + 3*i^2 -6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i, {i, 0, n-k}]/2, {n, 0, 10}, {k, 0, n}]] (* _G. C. Greubel_, Nov 02 2018 *)

%o (PARI) for(n=0,10, for(k=0,n, print1((-1)^(n-k)*sum(j=0, n-k, 2^j*binomial(n,j) *binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1)))/2, ", "))) \\ _G. C. Greubel_, Nov 02 2018

%o (Magma) [[(-1)^(n-k)*(&+[ 2^j*Binomial(n,j)*Binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1))/2: j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 02 2018

%o (GAP) T:=Flat(List([0..9],n->List([0..n],k->(-1)^(n-k)*Sum([0..n-k],i->Binomial(n,i)*Binomial(2*n-k-i,n)*(4-9*i+3*i^2-6*(i-1)*n+2*n^2)/((n-i+2)*(n-i+1))*2^i)/2))); # _Muniru A Asiru_, Nov 02 2018

%K sign,tabl

%O 0,2

%A _Paul Barry_, Aug 08 2006