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A123878 Product of signed and unsigned Morgan-Voyce triangles.

%I A123878 #9 Sep 08 2022 08:45:28
%S A123878 1,0,1,-1,0,1,-1,-3,0,1,0,-3,-5,0,1,1,3,-5,-7,0,1,1,9,10,-7,-9,0,1,0,
%T A123878 5,25,21,-9,-11,0,1,-1,-9,5,49,36,-11,-13,0,1,-1,-18,-50,-7,81,55,-13,
%U A123878 -15,0,1,0,-7,-70,-147,-39,121,78,-15,-17,0,1
%N A123878 Product of signed and unsigned Morgan-Voyce triangles.
%C A123878 Inverse is A123880.
%C A123878 Row sums are A123879.
%H A123878 G. C. Greubel, <a href="/A123878/b123878.txt">Rows n = 0..100 of triangle, flattened</a>
%F A123878 Riordan array ((1-x)/(1-x+x^2), x*(1-x)^2/(1-x+x^2)^2).
%F A123878 Number triangle: T(n,k) = Sum_{j=0..n} C(n+j,2*j)*C(j+k,2*k)*(-1)^(j-k).
%e A123878 Number triangle begins:
%e A123878    1;
%e A123878    0,  1;
%e A123878   -1,  0,  1;
%e A123878   -1, -3,  0,  1;
%e A123878    0, -3, -5,  0,  1;
%e A123878    1,  3, -5, -7,  0, 1;
%e A123878    1,  9, 10, -7, -9, 0, 1;
%t A123878 Table[Sum[(-1)^(j-k)*Binomial[n+j,2*j]*Binomial[j+k,2*k], {j,0,n}], {n, 0, 12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 08 2019 *)
%o A123878 (PARI) T(n,k) = sum(j=0,n, (-1)^(j-k)*binomial(n+j,2*j)*binomial(n+j,2*k) );
%o A123878 for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Aug 08 2019
%o A123878 (Magma) B:= Binomial; [(&+[(-1)^(j-k)*B(n+j,2*j)*B(n+j,2*k):j in [0..n]]) : k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 08 2019
%o A123878 (Sage) b=binomial; [[sum((-1)^(j-k)*b(n+j,2*j)*b(n+j,2*k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Aug 08 2019
%o A123878 (GAP) B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(j-k)*B(n+j,2*j)*B(n+j,2*k) )))); # _G. C. Greubel_, Aug 08 2019
%Y A123878 Cf. A085478, A123879, A123880.
%K A123878 easy,sign,tabl
%O A123878 0,8
%A A123878 _Paul Barry_, Oct 16 2006