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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified November 28 04:43 EST 2021. Contains 349400 sequences. (Running on oeis4.)