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A349815 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1 - x + x^2 + x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

%I A349815 #22 Dec 24 2021 13:10:22
%S A349815 1,-1,-1,1,1,-1,-2,-1,0,1,2,1,-1,-3,-4,-4,-2,2,4,4,3,1,-1,-4,-8,-12,
%T A349815 -13,-8,0,8,13,12,8,4,1,-1,-5,-13,-25,-37,-41,-33,-13,13,33,41,37,25,
%U A349815 13,5,1,-1,-6,-19,-44,-80,-116,-136,-124,-74,0,74,124,136,116,80,44,19,6,1,-1,-7,-26,-70,-149,-259,-376,-456,-450,-334,-124,124,334,450,456,376,259,149,70,26,7,1
%N A349815 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1 - x + x^2 + x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.
%C A349815 The row polynomials can be further factorized, since -3 - x + x^2 + 3*x^3 = -(1-x)*(1+x)^2 and 1 + x + x^2 + x^3 = (1+x)*(1+x^2).
%C A349815 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 [-1, -1, 1, 3] instead of [1, 1, 1, 1].
%C A349815 This is a companion to A008287 and A349813.
%H A349815 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 A349815 Triangle begins:
%e A349815    1;
%e A349815   -1, -1,   1,   1;
%e A349815   -1, -2,  -1,   0,   1,   2,   1;
%e A349815   -1, -3,  -4,  -4,  -2,   2,   4,   4,  3,  1;
%e A349815   -1, -4,  -8, -12, -13,  -8,   0,   8, 13, 12,  8,  4,  1;
%e A349815   -1, -5, -13, -25, -37, -41, -33, -13, 13, 33, 41, 37, 25, 13, 5, 1;
%e A349815   ...
%p A349815 t1:=-1-x+x^2+x^3;
%p A349815 m:=1+x+x^2+x^3;
%p A349815 lprint([3]);
%p A349815 for n from 1 to 12 do
%p A349815 w1:=expand(t1*m^(n-1));
%p A349815 w4:=series(w1,x,3*n+1);
%p A349815 w5:=seriestolist(w4);
%p A349815 lprint(w5);
%p A349815 od:
%Y A349815 Cf. A007318, A008287, A349812, A349813.
%Y A349815 The right half of the triangle gives A349816. For the central nonzero entries see A349818.
%K A349815 sign,tabf
%O A349815 0,7
%A A349815 _N. J. A. Sloane_, Dec 23 2021