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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.

%I A144823 #18 May 10 2019 11:45:58
%S A144823 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,
%T A144823 36,135,280,288,80,1,1,8,49,231,675,1168,864,168,1,1,9,64,364,1386,
%U A144823 3475,4672,2647,340,1,1,10,81,540,2548,8496,17375,18884,7968,698,1,1,11,100
%N 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.
%H A144823 Alois P. Heinz, <a href="/A144823/b144823.txt">Antidiagonals n = 1..100, flattened</a>
%H A144823 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%e A144823 Square array A(n,k) begins:
%e A144823    1,   1,    1,     1,     1,      1,      1,      1, ...
%e A144823    1,   1,    1,     1,     1,      1,      1,      1, ...
%e A144823    2,   3,    4,     5,     6,      7,      8,      9, ...
%e A144823    4,   9,   16,    25,    36,     49,     64,     81, ...
%e A144823    9,  30,   70,   135,   231,    364,    540,    765, ...
%e A144823   18,  90,  280,   675,  1386,   2548,   4320,   6885, ...
%e A144823   40, 288, 1168,  3475,  8496,  18130,  35008,  62613, ...
%e A144823   80, 864, 4672, 17375, 50976, 126910, 280064, 563517, ...
%p A144823 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);
%t A144823 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 *)
%Y A144823 Columns 1-9 give: A038044, A144817, A144316, A144818, A144819, A144820, A144317, A144821, A144822.
%Y A144823 Rows 1+2, 3-4 give: A000012, A000027, A000290, A002414.
%K A144823 eigen,nonn,tabl
%O A144823 1,6
%A A144823 _Alois P. Heinz_, Sep 21 2008