A144317 -id:A144317 - cp's OEIS frontend

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

Alois P. Heinz, Sep 17 2008

			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, ...

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

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).

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);

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 *)

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.

Original entry on oeis.org

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

Alois P. Heinz, Sep 21 2008

			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, ...

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

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

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

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);

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 *)

cp's OEIS Frontend

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.

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