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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. 4
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, -30, -31, -20, 0, 20, 31, 30, 20, 10, 3, -3, -13, -33, -63, -91, -101, -81, -31, 31, 81, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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).
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].
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:
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, -30, -31, -20, 0, 20, 31, 30, 20, 10, 3;
-3, -13, -33, -63, -91, -101, -81, -31, 31, 81, 101, 91, 63, 33, 13, 3;
...
MAPLE
t1:=-3-x+x^2+3*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
The right half of the triangle gives A349814.
Sequence in context: A301303 A285116 A356301 * A266529 A266509 A266539
KEYWORD
sign,tabf
AUTHOR
N. J. A. Sloane, Dec 23 2021
STATUS
approved

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Last modified March 29 05:28 EDT 2024. Contains 371264 sequences. (Running on oeis4.)