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.

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, -13, -8, 0, 8, 13, 12, 8, 4, 1, -1, -5, -13, -25, -37, -41, -33, -13, 13, 33, 41, 37, 25, 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

Offset: 0

Author

N. J. A. Sloane, Dec 23 2021

Keywords

• sign
• tabf

Comments

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).

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].

This is a companion to A008287 and A349813.

Examples

			Triangle begins:
   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, -13,  -8,   0,   8, 13, 12,  8,  4,  1;
  -1, -5, -13, -25, -37, -41, -33, -13, 13, 33, 41, 37, 25, 13, 5, 1;
  ...
		

Links

• Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]

Crossrefs

Cf. A007318, A008287, A349812, A349813.

The right half of the triangle gives A349816. For the central nonzero entries see A349818.

Programs

Maple

t1:=-1-x+x^2+x^3;
m:=1+x+x^2+x^3;
lprint([3]);
for n from 1 to 12 do
w1:=expand(t1*m^(n-1));
w4:=series(w1,x,3*n+1);
w5:=seriestolist(w4);
lprint(w5);
od: