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A349813 Triangle read by rows: row 1 is [3]; for n >= 1, row n gives coefficients of expansion of (-3 - x + x^2 + 3*x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

%I A349813 #26 Dec 24 2021 13:19:32
%S A349813 3,-3,-1,1,3,-3,-4,-3,0,3,4,3,-3,-7,-10,-10,-4,4,10,10,7,3,-3,-10,-20,
%T A349813 -30,-31,-20,0,20,31,30,20,10,3,-3,-13,-33,-63,-91,-101,-81,-31,31,81,
%U A349813 101,91,63,33,13,3,-3,-16,-49,-112,-200,-288,-336,-304,-182,0,182,304,336,288,200,112,49,16,3
%N A349813 Triangle read by rows: row 1 is [3]; for n >= 1, row n gives coefficients of expansion of (-3 - x + x^2 + 3*x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.
%C A349813 The row polynomials can be further factorized, since -3 - x + x^2 + 3*x^3 = -(1-x)*(3 + 4*x + 3*x^2) and 1 + x + x^2 + x^3 = (1+x)*(1+x^2).
%C A349813 The rule for constructing this triangle (ignoring row 0) is the same as that for A008287: each number is the sum of the four numbers immediately above it in the previous row. Here row 1 is [-3, -1, 1, 3] instead of [1, 1, 1, 1].
%H A349813 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 A349813 Triangle begins:
%e A349813    3;
%e A349813   -3,  -1,   1,   3;
%e A349813   -3,  -4,  -3,   0,   3,    4,   3;
%e A349813   -3,  -7, -10, -10,  -4,    4,  10,  10,  7,  3;
%e A349813   -3, -10, -20, -30, -31,  -20,   0,  20, 31, 30,  20, 10,  3;
%e A349813   -3, -13, -33, -63, -91, -101, -81, -31, 31, 81, 101, 91, 63, 33, 13, 3;
%e A349813   ...
%p A349813 t1:=-3-x+x^2+3*x^3;
%p A349813 m:=1+x+x^2+x^3;
%p A349813 lprint([3]);
%p A349813 for n from 1 to 12 do
%p A349813 w1:=expand(t1*m^(n-1));
%p A349813 w4:=series(w1,x,3*n+1);
%p A349813 w5:=seriestolist(w4);
%p A349813 lprint(w5);
%p A349813 od:
%Y A349813 Cf. A007318, A008287, A349812, A349815, A349819.
%Y A349813 The right half of the triangle gives A349814.
%K A349813 sign,tabf
%O A349813 0,1
%A A349813 _N. J. A. Sloane_, Dec 23 2021