A144324 - OEIS

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.

%I #15 May 10 2019 11:44:34

%S 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,

%T 1,1,64,1024,4216,4320,1168,80,1,1,128,4096,33264,67456,35008,4672,

%U 168,1,1,256,16384,264160,1064448,1083136,280064,18884,340,1,1,512,65536

%N 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.

%H Alois P. Heinz, <a href="/A144324/b144324.txt">Antidiagonals n = 1..65, flattened</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%e Square array begins:

%e 1, 1, 1, 1, 1, ...

%e 2, 4, 8, 16, 32, ...

%e 4, 16, 64, 256, 1024, ...

%e 9, 70, 540, 4216, 33264, ...

%p 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);

%t 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 *)

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

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

%K eigen,nonn,tabl

%O 1,6

%A _Alois P. Heinz_, Sep 17 2008