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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.
10
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
1,6
LINKS
N. J. A. Sloane, Transforms
EXAMPLE
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, ...
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 *)
CROSSREFS
Rows 1+2, 3-4 give: A000012, A000027, A000290, A002414.
Sequence in context: A138028 A009999 A322268 * A098446 A098447 A202784
KEYWORD
eigen,nonn,tabl
AUTHOR
Alois P. Heinz, Sep 21 2008
STATUS
approved