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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.
8
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, -12, -9, 0, 9, 12, 9, 4, 1, -1, -5, -14, -25, -30, -21, 0, 21, 30, 25, 14, 5, 1, -1, -6, -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
OFFSET
0,11
COMMENTS
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].
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, 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, -12, -9, 0, 9, 12, 9, 4, 1;
-1, -5, -14, -25, -30, -21, 0, 21, 30, 25, 14, 5, 1;
-1, -6, -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;
...
MAPLE
t1:=-1/x+x; m:=1/x+1+x;
lprint([1]);
for n from 1 to 12 do
w1:=expand(t1*m^(n-1));
w3:=expand(x^n*w1);
w4:=series(w3, x, 2*n+1);
w5:=seriestolist(w4);
lprint(w5);
od:
CROSSREFS
The left half of the triangle is A026300, the right half is A064189 (or A122896). The central (nonzero) column gives the Motzkin numbers A001006.
Sequence in context: A066518 A218491 A111165 * A029321 A029310 A348220
KEYWORD
sign,tabf
AUTHOR
N. J. A. Sloane, Dec 23 2021
STATUS
approved