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

0,3

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..500

Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions.

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

Example

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.

Prog

(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

Crossrefs

Cf. A001399, A098859.

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

Sequence in context: A082543 A050322 A325512 * A349149 A029012 A095699

Adjacent sequences: A211855 A211856 A211857 * A211859 A211860 A211861

Keyword

nonn,easy

Author

Matthew C. Russell, Apr 25 2012

Status

approved