A124840 Triangle, row sums = A008683, the Mobius sequence.

1, 1, -2, 1, -4, 2, 1, -6, 6, -1, 1, -8, 12, -4, -2, 1, -10, 20, -10, -10, 10, 1, -12, 30, -20, -30, 60, -30, 1, -14, 42, -35, -70, 210, -210, 76, 1, -16, 56, -56, -140, 560, -840, 608, -173, 1, -18, 72, -84, -252, 1260, -2520, 2736, -1557, 363, 1, -20, 90, -120, -420, 2520, -6300, 9120, -7785, 3630, -717

Offset: 1

Author

Gary W. Adamson, Nov 10 2006

Keywords

• sign
• tabl

Comments

Cf. A124839, the inverse binomial transform of mu(n), A008683.

Examples

			First few rows of the triangle are:
  1;
  1, -2;
  1, -4, 2;
  1, -6, 6, -1;
  1, -8, 12, -4, -2
  1, -10, 20, -10, -10, 10;
  1, -12, 30, -20, -30, 60, -30;
  ...
E.g., mu(5) = -1 = sum of row 5 terms: (1 - 8 + 12 - 4 - 2).
		

Crossrefs

Cf. A008683, A124839.

Programs

Mathematica

s[n_] := Sum[(-1)^(n-k) * Binomial[n-1, k-1] * MoebiusMu[k], {k, 1, n}]; row[n_] := Table[s[k+1] * Binomial[n, k], {k, 0, n}]; Array[row, 11, 0] // Flatten (* Amiram Eldar, Jun 01 2025 *)

Formula

Binomial transform of the diagonalized matrix using A124839; i.e., let A124839 (1, -2, 2, -1...) = the diagonal of an infinite matrix M; then the triangle (deleting the zeros) = P*M where P = Pascal's triangle as an infinite lower triangular matrix.

Extensions

a(39) corrected and more terms added by Amiram Eldar, Jun 01 2025