A211858 Number of partitions of n into parts <= 3 with the property that all parts have distinct multiplicities.

1, 1, 2, 2, 3, 4, 5, 7, 7, 8, 10, 14, 12, 19, 19, 19, 23, 30, 26, 37, 35, 37, 43, 52, 45, 60, 59, 61, 68, 80, 70, 90, 88, 91, 100, 113, 101, 126, 124, 127, 136, 153, 139, 168, 165, 168, 180, 199, 182, 216, 212, 216, 229, 251, 232, 269, 265, 270, 285, 309, 286

Offset: 0

Author

Matthew C. Russell, Apr 25 2012

Keywords

• nonn
• easy

Examples

			For n=3 the a(3)=2 partitions are {3} and {1,1,1}. Note that {2,1} does not count, as 1 and 2 appear with the same nonzero multiplicity.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..500
• Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions.

Crossrefs

Cf. A001399, A098859.

Cf. A105637, A211859, A211860, A211861, A211862, A211863.

Programs

Haskell

a211858 n = p 0 [] [1..3] n where
   p m ms _      0 = if m `elem` ms then 0 else 1
   p    []     _ = 0
   p m ms ks'@(k:ks) x
     | x < k       = 0
     | m == 0      = p 1 ms ks' (x - k) + p 0 ms ks x
     | m `elem` ms = p (m + 1) ms ks' (x - k)
     | otherwise   = p (m + 1) ms ks' (x - k) + p 0 (m : ms) ks x
-- Reinhard Zumkeller, Dec 27 2012

Formula

G.f.: -(2*x^17 +3*x^16 +5*x^15 +5*x^14 +4*x^13 +2*x^11 +2*x^9 +3*x^8 +5*x^7 +5*x^6 +6*x^5 +6*x^4 +5*x^3 +4*x^2 +2*x+1) / ((x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) *(x+1)^2 *(x^2+x+1)^2 *(x-1)^3). - Alois P. Heinz, Apr 26 2012