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%I #17 Feb 09 2017 10:39:56
%S 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,
%T 59,61,68,80,70,90,88,91,100,113,101,126,124,127,136,153,139,168,165,
%U 168,180,199,182,216,212,216,229,251,232,269,265,270,285,309,286
%N Number of partitions of n into parts <= 3 with the property that all parts have distinct multiplicities.
%H Reinhard Zumkeller, <a href="/A211858/b211858.txt">Table of n, a(n) for n = 0..500</a>
%H Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/dmp.html">Using generatingfunctionology to enumerate distinct-multiplicity partitions</a>.
%F 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
%e 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 (Haskell)
%o a211858 n = p 0 [] [1..3] n where
%o p m ms _ 0 = if m `elem` ms then 0 else 1
%o p _ _ [] _ = 0
%o p m ms ks'@(k:ks) x
%o | x < k = 0
%o | m == 0 = p 1 ms ks' (x - k) + p 0 ms ks x
%o | m `elem` ms = p (m + 1) ms ks' (x - k)
%o | otherwise = p (m + 1) ms ks' (x - k) + p 0 (m : ms) ks x
%o -- _Reinhard Zumkeller_, Dec 27 2012
%Y Cf. A001399, A098859.
%Y Cf. A105637, A211859, A211860, A211861, A211862, A211863.
%K nonn,easy
%O 0,3
%A _Matthew C. Russell_, Apr 25 2012
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