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

1,4

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.

Links

Alois P. Heinz, Rows n = 1..141, flattened

N. J. A. Sloane, Transforms

Example

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

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 *)

Crossrefs

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

Cf. A000012, A000005.

Sequence in context: A098050 A278984 A111579 * A144018 A258709 A239144

Adjacent sequences: A144371 A144372 A144373 * A144375 A144376 A144377

Keyword

eigen,nonn,tabl

Author

Alois P. Heinz, Sep 18 2008

Status

approved