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A123876 Riordan array (1/(1+2*x), x*(1+x)/(1+2*x)^2).

%I A123876 #12 Sep 08 2022 08:45:28
%S A123876 1,-2,1,4,-5,1,-8,18,-8,1,16,-56,41,-11,1,-32,160,-170,73,-14,1,64,
%T A123876 -432,620,-377,114,-17,1,-128,1120,-2072,1666,-704,164,-20,1,256,
%U A123876 -2816,6496,-6608,3649,-1178,223,-23,1,-512,6912,-19392,24192,-16722,7001,-1826,291,-26,1
%N A123876 Riordan array (1/(1+2*x), x*(1+x)/(1+2*x)^2).
%C A123876 Inverse of A116395.
%C A123876 Row sums are A123877.
%C A123876 Diagonal sums are (-1)^n*A085810(n).
%C A123876 Unsigned version is A114164.
%H A123876 G. C. Greubel, <a href="/A123876/b123876.txt">Rows n = 0..100 of triangle, flattened</a>
%F A123876 Number triangle T(n,k) = (-1)^(n-k)*Sum_{j=0..n} C(k,j-k)*C(n,j)*2^(n-j).
%F A123876 T(n,k) = T(n-1,k-1) - 4*T(n-1,k) + T(n-2,k-1) - 4*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = -2, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Jan 18 2014
%e A123876 Triangle begins
%e A123876     1;
%e A123876    -2,   1;
%e A123876     4,  -5,    1;
%e A123876    -8,  18,   -8,   1;
%e A123876    16, -56,   41, -11,   1;
%e A123876   -32, 160, -170,  73, -14, 1;
%t A123876 Table[(-1)^(n-k)*Sum[2^(n-j)*Binomial[k,j-k]*Binomial[n,j], {j,0,n}], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 08 2019 *)
%o A123876 (PARI) T(n,k) = b=binomial; (-1)^(n-k)*sum(j=0,n, 2^(n-j)*b(k,j-k)* b(n,j)); \\ _G. C. Greubel_, Aug 08 2019
%o A123876 (Magma) [(-1)^(n-k)*(&+[2^(n-j)*Binomial(k,j-k)*Binomial(n,j): j in [0..n]]): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 08 2019
%o A123876 (Sage)
%o A123876 b=binomial;
%o A123876 [[(-1)^(n-k)*sum(2^(n-j)*b(k,j-k)*b(n,j) for j in (0..n)) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Aug 08 2019
%o A123876 (GAP) Flat(List([0..12], n-> List([0..n], k-> (-1)^(n-k)*Sum([0..n], j-> 2^(n-j)*Binomial(k,j-k)*Binomial(n,j) ))));
%Y A123876 Cf. A085810, A114164, A116395, A123877.
%K A123876 easy,sign,tabl
%O A123876 0,2
%A A123876 _Paul Barry_, Oct 16 2006
%E A123876 More terms added by _G. C. Greubel_, Aug 08 2019