A211859 Number of partitions of n into parts <= 4 with the property that all parts have distinct multiplicities.

1, 1, 2, 2, 4, 4, 6, 8, 10, 10, 14, 18, 18, 26, 31, 30, 39, 48, 48, 61, 63, 73, 84, 101, 98, 124, 132, 147, 156, 188, 182, 223, 227, 257, 272, 322, 306, 367, 377, 417, 427, 499, 488, 564, 567, 645, 647, 740, 720, 828, 836, 920, 924, 1048, 1030, 1173, 1161

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

Formula

G.f.: (9*x^45 +20*x^44 +44*x^43 +76*x^42 +121*x^41 +172*x^40 +234*x^39 +292*x^38 +346*x^37 +380*x^36 +412*x^35 +415*x^34 +417*x^33 +401*x^32 +389*x^31 +365*x^30 +361*x^29 +351*x^28 +359*x^27 +365*x^26 +383*x^25 +391*x^24 +413*x^23 +422*x^22 +436*x^21 +444*x^20 +454*x^19 +454*x^18 +458*x^17 +450*x^16 +437*x^15 +415*x^14 +383*x^13 +342*x^12 +298*x^11 +248*x^10 +198*x^9 +152*x^8 +110*x^7 +76*x^6 +49*x^5 +30*x^4 +16*x^3 +8*x^2 +3*x +1) / ((x^2-x+1) *(x^4-x^3+x^2-x+1) *(x^6+x^3+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+x+1)^2 *(x^4+x^3+x^2+x+1)^2 *(x^2+1)^2 *(x+1)^3 *(x-1)^4). - Alois P. Heinz, Feb 09 2017

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)
a211859 n = p 0 [] [1..4] 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. A026810, A098859.

Cf. A105637, A211858, A211860, A211861, A211862, A211863.

Sequence in context: A334001 A341731 A183002 * A057601 A294150 A087135

Adjacent sequences: A211856 A211857 A211858 * A211860 A211861 A211862

Keyword

nonn,easy

Author

Matthew C. Russell, Apr 25 2012

Status

approved