A123878 - OEIS

%I #9 Sep 08 2022 08:45:28

%S 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 5,25,21,-9,-11,0,1,-1,-9,5,49,36,-11,-13,0,1,-1,-18,-50,-7,81,55,-13,

%U -15,0,1,0,-7,-70,-147,-39,121,78,-15,-17,0,1

%N Product of signed and unsigned Morgan-Voyce triangles.

%C Inverse is A123880.

%C Row sums are A123879.

%H G. C. Greubel, <a href="/A123878/b123878.txt">Rows n = 0..100 of triangle, flattened</a>

%F Riordan array ((1-x)/(1-x+x^2), x*(1-x)^2/(1-x+x^2)^2).

%F Number triangle: T(n,k) = Sum_{j=0..n} C(n+j,2*j)*C(j+k,2*k)*(-1)^(j-k).

%e Number triangle begins:

%e    1;

%e    0,  1;

%e   -1,  0,  1;

%e   -1, -3,  0,  1;

%e    0, -3, -5,  0,  1;

%e    1,  3, -5, -7,  0, 1;

%e    1,  9, 10, -7, -9, 0, 1;

%t 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 (PARI) T(n,k) = sum(j=0,n, (-1)^(j-k)*binomial(n+j,2*j)*binomial(n+j,2*k) );

%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Aug 08 2019

%o (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 (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 (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 Cf. A085478, A123879, A123880.

%K easy,sign,tabl

%O 0,8

%A _Paul Barry_, Oct 16 2006