A159934 Triangle, row sums = d(n) = A000005(n): M * Q; where M = an infinite lower Toeplitz matrix with A159933 in every column. Q = an infinite lower triangular matrix with d(n) shifted: (1, 1, 2, 2, 3, 2, 4, ...) as the main diagonal and the rest zeros.

1, 1, 1, -1, 1, 2, 0, -1, 2, 2, -1, 0, -2, 2, 3, 2, -1, 0, -2, 3, 2, -1, 2, -2, 0, -3, 2, 4, -1, -1, 4, -2, 0, -2, 4, 2, 3, -1, -2, 4, -3, 0, -4, 2, 4, -4, 3, -2, -2, 6, -2, 0, -2, 4, 3, 2, -4, 6, -2, -3, 4, -4, 0, -4, 3, 4

Offset: 1

Author

Gary W. Adamson, Apr 26 2009

Keywords

• tabl
• sign

Comments

Triangle = an infinite lower triangular Toeplitz matrix with the INVERTi transform of d(n) in every column; i.e., A159933: (1, 1, -1, 0, -1, 2, -1, ...). Row sums of the resulting eigentriangle of d(n) = d(n).

Sum of n-th row terms = rightmost term of next row.

Right border = d(n) shifted.

Examples

			First few rows of the triangle:
   1;
   1,  1;
  -1,  1,  2;
   0, -1,  2,  2;
  -1,  0, -2,  2,   3;
   2, -1,  0, -2,   3,  2;
  -1,  2, -2,  0,  -3,  2,   4;
  -1, -1,  4, -2,   0, -2,   4,  2;
   3, -1, -2,  4,  -3,  0,  -4,  2,  4;
  -4,  3, -2, -2,   6, -2,   0, -2,  4,  3;
   2, -4,  6, -2,  -3,  4,  -4,  0, -4,  3,  4;
   2,  2, -8,  6,  -3, -2,   8, -2,  0, -3,  4,  2;
  -3,  2,  4, -8,   9, -2,  -4,  4, -4,  0, -4,  2,  6;
   0, -3,  4,  4, -12,  6,  -4, -2,  8, -3,  0, -2,  6,  2;
   0,  0, -6,  4,   6, -8,  12, -2, -4,  6, -4,  0, -6,  2, 4;
   6,  0,  0, -6,   6,  4, -16,  6, -4, -3,  8, -2,  0, -2, 4, 4;
  ...
Example: row 6 = (2, -1, 0, -2, 3, 2) = termwise products of (2, -1, 0, -1, 1, 1) and (1, 1, 2, 2, 3, 2); with dot product sum = 4 = d(6).
		

Crossrefs

Cf. A000005, A159933.