A144823 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Dirichlet convolution with a_k (DC:(b,a_k)->a) applied k times.

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 30, 18, 1, 1, 6, 25, 70, 90, 40, 1, 1, 7, 36, 135, 280, 288, 80, 1, 1, 8, 49, 231, 675, 1168, 864, 168, 1, 1, 9, 64, 364, 1386, 3475, 4672, 2647, 340, 1, 1, 10, 81, 540, 2548, 8496, 17375, 18884, 7968, 698, 1, 1, 11, 100

Offset: 1

Author

Alois P. Heinz, Sep 21 2008

Keywords

• eigen
• nonn
• tabl

Examples

			Square array A(n,k) begins:
   1,   1,    1,     1,     1,      1,      1,      1, ...
   1,   1,    1,     1,     1,      1,      1,      1, ...
   2,   3,    4,     5,     6,      7,      8,      9, ...
   4,   9,   16,    25,    36,     49,     64,     81, ...
   9,  30,   70,   135,   231,    364,    540,    765, ...
  18,  90,  280,   675,  1386,   2548,   4320,   6885, ...
  40, 288, 1168,  3475,  8496,  18130,  35008,  62613, ...
  80, 864, 4672, 17375, 50976, 126910, 280064, 563517, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..100, flattened
• N. J. A. Sloane, Transforms

Crossrefs

Columns 1-9 give: A038044, A144817, A144316, A144818, A144819, A144820, A144317, A144821, A144822.

Rows 1+2, 3-4 give: A000012, A000027, A000290, A002414.

Programs

Maple

with(numtheory): dc:= proc(b,c) proc(n) option remember; add(b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

Mathematica

dc[b_, c_] := Module[{proc}, proc[n_] := proc[n] = Sum [b[d] *c[n/d], {d, If[n < 0, {}, Divisors[n]]}]; proc]; A [n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], a]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[Table [A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)