A048600 Array a(n,k) = number of colors that can be produced by n units of paint from k primary colors, read by descending antidiagonals.

1, 1, 2, 1, 3, 3, 1, 5, 6, 4, 1, 7, 13, 10, 5, 1, 11, 22, 26, 15, 6, 1, 13, 40, 51, 45, 21, 7, 1, 19, 55, 103, 100, 71, 28, 8, 1, 23, 88, 161, 221, 176, 105, 36, 9, 1, 29, 118, 277, 386, 422, 287, 148, 45, 10

Offset: 1

Author

Jurjen N.E. Bos

Keywords

• easy
• nice
• nonn
• tabl

Examples

			Table array begins:
  1  1  1   1   1
  2  3  5   7  11
  3  6 13  22  40
  4 10 26  51 103
  5 15 45 100 221
  ...
a(3,2) = 6 because you can take each color once, or mix two colors.
		

Crossrefs

Cf. A005728 (row 2), A048134 (row 3). Cf. A048240, A048241.

Programs

Mathematica

max = 10; col[k_] := Accumulate[ Table[ Sum[ MoebiusMu[n/d]*Product[d+j, {j, 1, k}]/k!, {d, Divisors[n]}], {n, 1, max}]]; t = Table[col[k], {k, 0, max-1}] // Transpose; Flatten[ Table[ t[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Dec 26 2012 *)

Formula

All partitions of size n: if GCD is not 1, skip; else: fill the partition with zeros to get k numbers; count occurrences of each number (e.g.: 2 2 1 0 0 0 becomes 2 1 3); compute multinomial of k over these digits (e.g. 2 1 3 becomes 6!/(2!*1!*3!) = 60); sum.

Extensions

Name edited by Michel Marcus, Aug 11 2024