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

%I A211859 #16 Feb 09 2017 11:14:46
%S A211859 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,
%T A211859 124,132,147,156,188,182,223,227,257,272,322,306,367,377,417,427,499,
%U A211859 488,564,567,645,647,740,720,828,836,920,924,1048,1030,1173,1161
%N A211859 Number of partitions of n into parts <= 4 with the property that all parts have distinct multiplicities.
%H A211859 Alois P. Heinz, <a href="/A211859/b211859.txt">Table of n, a(n) for n = 0..1000</a>
%H A211859 Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/dmp.html">Using generatingfunctionology to enumerate distinct-multiplicity partitions</a>.
%F A211859 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
%e A211859 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.
%o A211859 (Haskell)
%o A211859 a211859 n = p 0 [] [1..4] n where
%o A211859    p m ms _      0 = if m `elem` ms then 0 else 1
%o A211859    p _ _  []     _ = 0
%o A211859    p m ms ks'@(k:ks) x
%o A211859      | x < k       = 0
%o A211859      | m == 0      = p 1 ms ks' (x - k) + p 0 ms ks x
%o A211859      | m `elem` ms = p (m + 1) ms ks' (x - k)
%o A211859      | otherwise   = p (m + 1) ms ks' (x - k) + p 0 (m : ms) ks x
%o A211859 -- _Reinhard Zumkeller_, Dec 27 2012
%Y A211859 Cf. A026810, A098859.
%Y A211859 Cf. A105637, A211858, A211860, A211861, A211862, A211863.
%K A211859 nonn,easy
%O A211859 0,3
%A A211859 _Matthew C. Russell_, Apr 25 2012