A357914 Iterated partial sums of the Moebius mu function, square array read by ascending antidiagonals.

1, 1, -1, 1, 0, -1, 1, 1, -1, 0, 1, 2, 0, -1, -1, 1, 3, 2, -1, -2, 1, 1, 4, 5, 1, -3, -1, -1, 1, 5, 9, 6, -2, -4, -2, 0, 1, 6, 14, 15, 4, -6, -6, -2, 0, 1, 7, 20, 29, 19, -2, -12, -8, -2, 1, 1, 8, 27, 49, 48, 17, -14, -20, -10, -1, -1, 1, 9, 35, 76, 97, 65, 3, -34, -30, -11, -2, 0

Offset

1,12

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)

Formula

T(1,k) = A008683(k) for k >= 1; T(n,k) = Sum_{i=1..k} T(n-1,i) for n > 1, k >= 1.

Example

Array begins:
  n\k|  1   2   3    4    5    6     7     8     9    10  ...
  ---+-------------------------------------------------------
   1 |  1, -1, -1,   0,  -1,   1,   -1,    0,    0,    1, ... = A008683
   2 |  1,  0, -1,  -1,  -2,  -1,   -2,   -2,   -2,   -1, ... = A002321
   3 |  1,  1,  0,  -1,  -3,  -4,   -6,   -8,  -10,  -11, ... = A091555
   4 |  1,  2,  2,   1,  -2,  -6,  -12,  -20,  -30,  -41, ...
   5 |  1,  3,  5,   6,   4,  -2,  -14,  -34,  -64, -105, ...
   6 |  1,  4,  9,  15,  19,  17,    3,  -31,  -95, -200, ...
   7 |  1,  5, 14,  29,  48,  65,   68,   37,  -58, -258, ...
   8 |  1,  6, 20,  49,  97, 162,  230,  267,  209,  -49, ...
   9 |  1,  7, 27,  76, 173, 335,  565,  832, 1041,  992, ...
  10 |  1,  8, 35, 111, 284, 619, 1184, 2016, 3057, 4049, ...
  ...

Mathematica

A357914list[dmax_]:=With[{a=Reverse[NestList[Accumulate[Most[#]]&, MoebiusMu[Range[dmax]], dmax-1]]}, Array[Diagonal[a, #]&, dmax, 1-dmax]];
A357914list[10] (* Generates 10 antidiagonals *)

Crossrefs

Cf. A008683 (row 1), A002321 (row 2), A091555 (row 3), A000012 (column 1), A368429 (main diagonal).

Discarding terms above the main diagonal: A001477 (column 2), A000096 (column 3), A005286 (column 4).

Sequence in context: A175358 A235330 A029394 * A035467 A254045 A024996

Adjacent sequences: A357911 A357912 A357913 * A357915 A357916 A357917

Keyword

sign,tabl

Author

Paolo Xausa, Jan 18 2023

Status

approved