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A048600 Triangle a(n,k) = number of colors that can be produced by n units of paint from k primary colors. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified June 26 01:07 EDT 2017. Contains 288748 sequences.