A144324 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 (DC:(b,b)->a) applied k times.

1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 16, 9, 1, 1, 16, 64, 70, 18, 1, 1, 32, 256, 540, 280, 40, 1, 1, 64, 1024, 4216, 4320, 1168, 80, 1, 1, 128, 4096, 33264, 67456, 35008, 4672, 168, 1, 1, 256, 16384, 264160, 1064448, 1083136, 280064, 18884, 340, 1, 1, 512, 65536

Offset: 1

Author

Alois P. Heinz, Sep 17 2008

Keywords

• eigen
• nonn
• tabl

Examples

			Square array begins:
  1,  1,   1,    1,     1, ...
  1,  1,   1,    1,     1, ...
  2,  4,   8,   16,    32, ...
  4, 16,  64,  256,  1024, ...
  9, 70, 540, 4216, 33264, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..65, flattened
• N. J. A. Sloane, Transforms

Crossrefs

Columns 1-9 give: A038044, A144316, A144317, A144318, A144319, A144320, A144321, A144322, A144323.

Rows 1+2, 3-4 give: A000012, A000079(k+1), A000302(k+1).

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],b[t-1]) 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..11);

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], b[t-1]]]; 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, 11}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)