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

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

1,3

Links

Table of n, a(n) for n=1..55.

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.

Example

Table read by antidiagonals: (1), (1 2), (1 3 3), (1 5 6 4), (1 7 13 10 5), ...
a(2,3)=6 because you can take each color once, or mix two colors.

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 *)

Crossrefs

Row 2 = A005728, row 3 = A048134. Cf. A048240, A048241.

Sequence in context: A257243 A097351 A207330 * A100578 A061315 A144265

Adjacent sequences: A048597 A048598 A048599 * A048601 A048602 A048603

Keyword

easy,nice,nonn,tabl

Author

Jurjen N.E. Bos

Status

approved