A357914 - OEIS

%I #31 Dec 29 2023 13:53:24

%S 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,

%T 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,

%U 1,8,27,49,48,17,-14,-20,-10,-1,-1,1,9,35,76,97,65,3,-34,-30,-11,-2,0

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

%H Paolo Xausa, <a href="/A357914/b357914.txt">Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)</a>

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

%e Array begins:

%e   n\k|  1   2   3    4    5    6     7     8     9    10  ...

%e   ---+-------------------------------------------------------

%e    1 |  1, -1, -1,   0,  -1,   1,   -1,    0,    0,    1, ... = A008683

%e    2 |  1,  0, -1,  -1,  -2,  -1,   -2,   -2,   -2,   -1, ... = A002321

%e    3 |  1,  1,  0,  -1,  -3,  -4,   -6,   -8,  -10,  -11, ... = A091555

%e    4 |  1,  2,  2,   1,  -2,  -6,  -12,  -20,  -30,  -41, ...

%e    5 |  1,  3,  5,   6,   4,  -2,  -14,  -34,  -64, -105, ...

%e    6 |  1,  4,  9,  15,  19,  17,    3,  -31,  -95, -200, ...

%e    7 |  1,  5, 14,  29,  48,  65,   68,   37,  -58, -258, ...

%e    8 |  1,  6, 20,  49,  97, 162,  230,  267,  209,  -49, ...

%e    9 |  1,  7, 27,  76, 173, 335,  565,  832, 1041,  992, ...

%e   10 |  1,  8, 35, 111, 284, 619, 1184, 2016, 3057, 4049, ...

%e   ...

%t A357914list[dmax_]:=With[{a=Reverse[NestList[Accumulate[Most[#]]&, MoebiusMu[Range[dmax]], dmax-1]]}, Array[Diagonal[a, #]&, dmax, 1-dmax]];

%t A357914list[10] (* Generates 10 antidiagonals *)

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

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

%K sign,tabl

%O 1,12

%A _Paolo Xausa_, Jan 18 2023