A144082 Eigentriangle generated from inverse of 6th cyclotomic polynomial, row sums = n+1.

1, 1, 1, 0, 1, 2, -1, 0, 2, 3, -1, -1, 0, 3, 4, 0, -1, -2, 0, 4, 5, 1, 0, -2, -3, 0, 5, 6, 1, 1, 0, -3, -4, 0, 6, 7, 0, 1, 2, 0, -4, -5, 0, 7, 8, -1, 0, 2, 3, 0, -5, -6, 0, 8, 9, -1, -1, 0, 3, 4, 0, -6, -7, 0, 9, 10, 0, -1, -2, 0, 4, 5, 0, -7, -8, 0, 10, 11, 1, 0, -2, -3, 0, 5, 6, 0, -8, -9, 0

Offset

0,6

Comments

Left border = A010892. Right border = (1, 1, 2, 3, 4,...), row sums = (1, 2, 3,...).

Links

Table of n, a(n) for n=0..88.

Formula

Eigentriangle by rows, T(n,k) = A010892(n-k)*A000027(k-1). A010892 = the inverse of the 6th cyclotomic polynomial: (1, 1, 0, -1, -1, 0,...); and A000027(k-1) = (1, 2, 3,...) offset, = (1, 1, 2, 3,...).

Let A = an infinite lower triangular decrescendo subsequences matrix with A010892: (1, 1, 0, -1, -1, 0,...) in every column: (1; 1,1; 0,1,1; -1,0,1,1;...); let B = an infinite lower triangular matrix with (1,1,2,3,...) in the main diagonal and the rest zeros; then A144082 = A*B.

Example

First few rows of the triangle =
1;
1, 1;
0, 1, 2;
-1, 0, 2, 3;
-1, -1, 0, 3, 4;
0, -1, -2, 0, 4, 5;
1, 0, -2, -3, 0, 5, 6;
1, 1, 0, -3, -4, 0, 6, 7;
0, 1, 2, 0, -4, -5, 0, 7, 8;
-1, 0, 2, 3, 0, -5, -6, 0, 8, 9;
-1, -1, 0, 3, 4, -6, -7, 0, 9, 10;
0, -1, -2, 0, 4, 5, 0, -7, -8, 0, 10, 11;
1, 0, -2, -3, 0, 5, 6, 0, -8, -9, 0, 11, 12;
...
Example: row 3 = (-1, 0, 3, 4) = termwise product of (-1, 0, 1, 1) and (1, 2, 3, 4) = (-1*1, 0*2, 1*3, 1*4). (-1, 0, 1, 1) = first 4 terms of A010892 reversed.

Crossrefs

Cf. A000027, A010892.

Sequence in context: A356027 A123949 A236358 * A145579 A167655 A346148

Adjacent sequences: A144079 A144080 A144081 * A144083 A144084 A144085

Keyword

tabl,sign

Author

Gary W. Adamson, Sep 10 2008

Status

approved