A171978 Number of partitions of n into fractions k/(k+1), 0 < k <= n.

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, 394553026, 799614318, 3108061596, 5172450911, 21360507716, 35407697816, 81523452326, 238510777299

Offset: 0

Author

Reinhard Zumkeller, Jan 20 2010

Keywords

• nonn

Examples

			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.
		

Links

• Index entries for sequences related to partitions

Programs

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

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 *)

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).

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

a(25)-a(32) from Jinyuan Wang, Dec 13 2024