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A349812 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.

%I A349812 #24 Dec 24 2021 13:27:03
%S A349812 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 A349812 -12,-9,0,9,12,9,4,1,-1,-5,-14,-25,-30,-21,0,21,30,25,14,5,1,-1,-6,
%U A349812 -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 A349812 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 A349812 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 A349812 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 A349812 Triangle begins:
%e A349812    1;
%e A349812   -1,  0,   1;
%e A349812   -1, -1,   0,   1,    1;
%e A349812   -1, -2,  -2,   0,    2,    2,    1;
%e A349812   -1, -3,  -5,  -4,    0,    4,    5,    3,  1;
%e A349812   -1, -4,  -9, -12,   -9,    0,    9,   12,  9,   4,   1;
%e A349812   -1, -5, -14, -25,  -30,  -21,    0,   21, 30,  25,  14,   5,   1;
%e A349812   -1, -6, -20, -44,  -69,  -76,  -51,    0, 51,  76,  69,  44,  20,  6,  1;
%e A349812   -1, -7, -27, -70, -133, -189, -196, -127,  0, 127, 196, 189, 133, 70, 27, 7, 1;
%e A349812   ...
%p A349812 t1:=-1/x+x; m:=1/x+1+x;
%p A349812 lprint([1]);
%p A349812 for n from 1 to 12 do
%p A349812 w1:=expand(t1*m^(n-1));
%p A349812 w3:=expand(x^n*w1);
%p A349812 w4:=series(w3,x,2*n+1);
%p A349812 w5:=seriestolist(w4);
%p A349812 lprint(w5);
%p A349812 od:
%Y A349812 Cf. A007318, A027907, A112467, A349813, A348815.
%Y A349812 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 A349812 sign,tabf
%O A349812 0,11
%A A349812 _N. J. A. Sloane_, Dec 23 2021