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 A171978 Number of partitions of n into fractions k/(k+1), 0
 1, 1, 2, 4, 7, 22, 37, 84, 172, 454, 745, 2904, 4722, 10464, 38546, 88769, 147439, 475153, 785894, 3140342, 10412267, 19169464, 32132160, 125087460, 224341028 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS FORMULA a(n) = q(n,1) with q(x,k) = if x < k/(k+1) then 0^x else if k>n then 0 else q(x-k/(k+1),k) + q(x,k+1). EXAMPLE a(3) = 4 partitions into parts 1/2, 2/3, or 3/4: #1: 3/4 + 3/4 + 3/4 + 3/4 = 3, #2: (3/4 + 3/4) + (1/2 + 1/2 + 1/2) = 3, #3: (2/3 + 2/3 + 2/3) + (1/2 + 1/2) = 3, #4: 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 3; a(4) = 7 partitions into parts 1/2, 2/3, 3/4, or 4/5: #1: 4/5 + 4/5 + 4/5 + 4/5 + 4/5 = 4, #2: (3/4 + 3/4 + 3/4 + 3/4) + (1/2 + 1/2) = 4, #3: (3/4 + 3/4) + (2/3 + 2/3 + 2/3) + 1/2 = 4, #4: (3/4 + 3/4) + (1/2 + 1/2 + 1/2 + 1/2 + 1/2) = 4, #5: 2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 4, #6: 2/3 + 2/3 + 2/3 + 1/2 + 1/2 + 1/2 + 1/2 = 4, #7: 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 4. MAPLE b:= proc(n, k) option remember;       `if`(n=0, 1, `if`(k=0 or isprime(k+2) and irem(denom(n),        k+2)=0, 0, b(n, k-1)+`if`(k>k*n+n, 0, b(n-k/(k+1), k))))     end: a:= n-> b(n, n): seq(a(n), n=0..16);  # Alois P. Heinz, Jul 18 2012 MATHEMATICA b[n_, k_] := b[n, k] = If[n==0, 1, If[k==0 || PrimeQ[k+2] && Mod[ Denominator[n], k+2]==0, 0, b[n, k-1] + If[k>k*n+n, 0, b[n-k/(k+1), k]]] ]; a[n_] := b[n, n]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *) PROG (Haskell) -- import Data.Ratio ((%)) a171978 n = q (fromInteger n) \$ zipWith (%) [1..n] [2..] where    q 0 _         = 1    q _ []        = 0    q x ks'@(k:ks)      | x < k     = fromEnum (x == 0)      | otherwise = q (x - k) ks' + q x ks -- Reinhard Zumkeller, Apr 01 2012 CROSSREFS Sequence in context: A102984 A103017 A091833 * A290571 A026080 A071795 Adjacent sequences:  A171975 A171976 A171977 * A171979 A171980 A171981 KEYWORD more,nonn AUTHOR Reinhard Zumkeller, Jan 20 2010 EXTENSIONS Offset corrected and a(16) added by Reinhard Zumkeller, Apr 01 2012 a(17)-a(23) from Alois P. Heinz, Jul 18 2012 a(24) from Alois P. Heinz, Sep 27 2014 STATUS approved

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Last modified April 14 21:21 EDT 2021. Contains 342962 sequences. (Running on oeis4.)