A144374 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k begins with (k+1) 1's and a_k(n) shifts k places down under Dirichlet convolution.

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 9, 2, 2, 1, 1, 18, 5, 2, 2, 1, 1, 40, 4, 3, 2, 2, 1, 1, 80, 12, 4, 3, 2, 2, 1, 1, 168, 8, 6, 2, 3, 2, 2, 1, 1, 340, 28, 6, 6, 2, 3, 2, 2, 1, 1, 698, 17, 10, 4, 4, 2, 3, 2, 2, 1, 1, 1396, 60, 13, 8, 4, 4, 2, 3, 2, 2, 1, 1, 2844, 34, 16, 5, 6, 2, 4, 2, 3, 2, 2, 1, 1, 5688

Offset: 1

Author

Alois P. Heinz, Sep 18 2008

Keywords

• eigen
• nonn
• tabl

Comments

Sequence a_k of column k begins with k terms from A000012 (only the last is in the triangle), followed by the first (k+1) terms from A000005.

Examples

			Triangle begins:
   1;
   1,  1;
   2,  1, 1;
   4,  2, 1, 1;
   9,  2, 2, 1, 1;
  18,  5, 2, 2, 1, 1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• N. J. A. Sloane, Transforms

Crossrefs

Columns 1-9 give: A038044, A144366, A144367, A144368, A144369, A144370, A144371, A144372, A144373.

Cf. A000012, A000005.

Programs

Maple

with(numtheory): dck:= proc(b,c) proc(n, k) option remember; add(b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: seq(seq(T(n, k), k=1..n), n=1..14);

Mathematica

dck[b_, c_][n_, k_] := dck[b, c][n, k] = Sum[b[d, k]*c[n/d, k], {d, If[n < 0, {}, Divisors[n]]}]; B = dck[T, T]; T[n_, k_] := If[n <= k, 1, B[n-k, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)