OFFSET
0,7
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].
LINKS
Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]
EXAMPLE
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;
...
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:
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
N. J. A. Sloane, Dec 23 2021
STATUS
approved