A209815 Number of partitions of 2n in which every part is <n; also, the number of partitions of 2 into rational numbers a/b such that 0<a<b<=n and b divides n.

0, 1, 4, 10, 23, 47, 90, 164, 288, 488, 807, 1303, 2063, 3210, 4920, 7434, 11098, 16380, 23928, 34624, 49668, 70667, 99795, 139935, 194930, 269857, 371413, 508363, 692195, 937838, 1264685, 1697810, 2269557, 3021462, 4006812, 5293650, 6968730, 9142306, 11954194

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

a(n) = A008284(3*n-1,n-1). - Hans Loeblich Apr 18 2019

Example

The 4 partitions of 6 with parts <3:
2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1.
Matching partitions of 2 into rationals as described:
2/3 + 2/3 + 2/3
2/3 + 2/3 + 1/3 + 1/3
2/3 + 1/3 + 1/3 + 1/3 + 1/3
1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(2*n, n-1):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 09 2012

Mathematica

f[n_] := Length[Select[IntegerPartitions[2 n], First[#] <= n - 1 &]];  Table[f[n], {n, 1, 34}]  (* A209815 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[2*n, n-1]; Table [a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

Prog

(Haskell)
a209815 n = p [1..n-1] (2*n) where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 14 2013

Crossrefs

Cf. A209816.

Cf. A231429.

Sequence in context: A305102 A008268 A084446 * A158671 A001980 A266376

Adjacent sequences: A209812 A209813 A209814 * A209816 A209817 A209818

Keyword

nonn

Author

Clark Kimberling, Mar 13 2012

Extensions

More terms from Alois P. Heinz, Jul 09 2012

Status

approved