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A112475 Riordan array (1/(1+x),x(1+x)/(1-x)).

%I A112475 #15 May 26 2024 08:24:48
%S A112475 1,-1,1,1,1,1,-1,1,3,1,1,1,5,5,1,-1,1,7,13,7,1,1,1,9,25,25,9,1,-1,1,
%T A112475 11,41,63,41,11,1,1,1,13,61,129,129,61,13,1,-1,1,15,85,231,321,231,85,
%U A112475 15,1,1,1,17,113,377,681,681,377,113,17,1
%N A112475 Riordan array (1/(1+x),x(1+x)/(1-x)).
%C A112475 Equivalent to Delannoy triangle A008288 with prepended column 1,-1,1,-1,... Row sums are A111954. Diagonal sums are A112476. Inverse is A112477.
%F A112475 T(n,k) = Sum{j=0..n-k} C(k-1,j)*C(n-j-1,n-k-j).
%e A112475 Triangle starts:
%e A112475    1;
%e A112475   -1, 1;
%e A112475    1, 1, 1;
%e A112475   -1, 1, 3,  1;
%e A112475    1, 1, 5,  5,  1;
%e A112475   -1, 1, 7, 13,  7, 1;
%e A112475    1, 1, 9, 25, 25, 9, 1;
%e A112475   ...
%t A112475 T[n_,k_]:=SeriesCoefficient[(x(1+x)/(1-x))^k/(1+x),{x,0,n}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* _Stefano Spezia_, May 26 2024 *)
%Y A112475 Cf. A008288, A111954, A112476, A112477.
%K A112475 easy,sign,tabl
%O A112475 0,9
%A A112475 _Paul Barry_, Sep 07 2005