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%I #24 Dec 24 2021 13:27:03
%S 1,-1,0,1,-1,-1,0,1,1,-1,-2,-2,0,2,2,1,-1,-3,-5,-4,0,4,5,3,1,-1,-4,-9,
%T -12,-9,0,9,12,9,4,1,-1,-5,-14,-25,-30,-21,0,21,30,25,14,5,1,-1,-6,
%U -20,-44,-69,-76,-51,0,51,76,69,44,20,6,1,-1,-7,-27,-70,-133,-189,-196,-127,0,127,196,189,133,70,27,7,1
%N Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1/x + x)*(1/x + 1 + x)^(n-1) in order of increasing powers of x.
%C The rule for constructing this triangle (ignoring row 0) is the same as that for A027907: each number is the sum of the three numbers immediately above it in the previous row. Here row 1 is [-1, 0, 1] instead of [1, 1, 1].
%H Jack Ramsay, <a href="/A349812/a349812.pdf">On Arithmetical Triangles</a>, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]
%e Triangle begins:
%e 1;
%e -1, 0, 1;
%e -1, -1, 0, 1, 1;
%e -1, -2, -2, 0, 2, 2, 1;
%e -1, -3, -5, -4, 0, 4, 5, 3, 1;
%e -1, -4, -9, -12, -9, 0, 9, 12, 9, 4, 1;
%e -1, -5, -14, -25, -30, -21, 0, 21, 30, 25, 14, 5, 1;
%e -1, -6, -20, -44, -69, -76, -51, 0, 51, 76, 69, 44, 20, 6, 1;
%e -1, -7, -27, -70, -133, -189, -196, -127, 0, 127, 196, 189, 133, 70, 27, 7, 1;
%e ...
%p t1:=-1/x+x; m:=1/x+1+x;
%p lprint([1]);
%p for n from 1 to 12 do
%p w1:=expand(t1*m^(n-1));
%p w3:=expand(x^n*w1);
%p w4:=series(w3,x,2*n+1);
%p w5:=seriestolist(w4);
%p lprint(w5);
%p od:
%Y Cf. A007318, A027907, A112467, A349813, A348815.
%Y The left half of the triangle is A026300, the right half is A064189 (or A122896). The central (nonzero) column gives the Motzkin numbers A001006.
%K sign,tabf
%O 0,11
%A _N. J. A. Sloane_, Dec 23 2021
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