A144871 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where sequence a_k of column k is shadow transform of C(n+k-1,k).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 3, 1, 3, 4, 2, 2, 1, 1, 1, 1, 1, 2, 3, 5, 1, 1, 2, 1, 1, 1, 1, 1, 3, 4, 6, 2, 2, 4, 2, 1, 1, 1, 2, 1, 4, 1, 1, 1, 4, 4, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 4, 3, 2, 1

Offset: 1

Author

Alois P. Heinz, Sep 23 2008

Keywords

• nonn
• tabl
• look

Comments

Row sequences have periods 1, 1, 3, 8, ... given in A144872.

Examples

			Square array begins:
  1, 1, 1, 1, 1, 1, ...
  1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 1, 2, 1, ...
  1, 1, 2, 1, 1, 2, ...
  1, 2, 3, 4, 1, 1, ...
  1, 2, 1, 1, 3, 3, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..141, flattened
• Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.
• OEIS Wiki, Shadow transform.
• N. J. A. Sloane, Transforms.

Crossrefs

Columns 1-9 give: A000012, A068068, A072457, A144865, A144866, A144867, A144868, A144869, A144870.

Rows 1+2, 3 give: A000012, A101825.

Periods of the row sequences: A144872.

Cf. A007318.

Programs

Maple

shadow:= proc(p) proc(n) add(`if`(modp(p(j), n)=0, 1, 0), j=0..n-1)
         end end:
f:= proc(k) proc(n) binomial(n+k-1, k) end end:
A:= (n, k)-> shadow(f(k))(n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..20);

Mathematica

shadow[p_] := Function[n, Sum[If[Mod[p[j], n] == 0, 1, 0], {j, 0, n-1}]]; f[k_] := Function[n, Binomial[n+k-1, k]]; a[n_, k_] := shadow[f[k]][n]; Table[Table[a[n, 1+d-n], {n, 1, d}], {d, 1, 20}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)