A124839 -id:A124839 - cp's OEIS frontend

A104688 Binomial transform of Moebius sequence.

1, 0, -2, -5, -10, -18, -30, -48, -77, -127, -213, -351, -551, -817, -1181, -1819, -3304, -7003, -15454, -32185, -59830, -94733, -116204, -70931, 138782, 634477, 1440741, 2129014, 995270, -6559829, -30802323, -91672920, -223074852, -473661244, -893720326, -1483495634, -2049478628

Offset: 1

Gary W. Adamson, Apr 26 2005

sign

			G.f.: A(x) = x - 2*x^3 - 5*x^4 - 10*x^5 - 18*x^6 - 30*x^7 - 48*x^8 - 77*x^9 - 127*x^10 - 213*x^11 - 351*x^12 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms

Cf. A008683, A124839.

Table[Sum[Binomial[n-1, k-1]*MoebiusMu[k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Jun 01 2025 *)

G.f. A(x) satisfies x = Sum_{n>=1} A( x^n/(1 + x^n) ). - Paul D. Hanna, Jun 03 2025

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

Original entry on oeis.org

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

Gary W. Adamson, Nov 10 2006

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

			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).

Cf. A008683, A124839.

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 *)

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.

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

cp's OEIS Frontend

A104688 Binomial transform of Moebius sequence.

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