A079122 Number of ways to partition 2*n into distinct positive integers not greater than n.

1, 0, 0, 1, 1, 3, 5, 8, 13, 21, 31, 46, 67, 95, 134, 186, 253, 343, 461, 611, 806, 1055, 1369, 1768, 2270, 2896, 3678, 4649, 5847, 7325, 9141, 11359, 14069, 17367, 21363, 26202, 32042, 39068, 47512, 57632, 69728, 84167, 101365, 121801, 146053, 174777

Offset

0,6

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..80 from Reinhard Zumkeller)

Formula

a(n) = b(0, n), b(m, n) = 1 + sum(b(i, j): m<i<j<n & i+j=2*n).

Coefficient of x^(2*n) in Product_{k=1..n} (1+x^k). - Vladeta Jovovic, Aug 07 2003

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 22 2015

Example

a(4)=1 [1+3+4=2*4]; a(5)=3 [1+2+3+4=1+4+5=2+3+5=2*5].

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-1))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 18 2013

Mathematica

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@ #] == 1 &]; Table[d[n], {n, 1, 12}]
TableForm[%]
f[n_] := Length[Select[d[2 n], First[#] <= n &]]
Table[f[n], {n, 1, 20}]  (* A079122 *)
(* Clark Kimberling, Mar 13 2012 *)
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-1]]]]; a[n_] := b[2*n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)
Table[SeriesCoefficient[Product[1 + x^(k/2), {k, 1, n}], {x, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Jan 16 2024 *)

Prog

(Haskell)
a079122 n = p [1..n] (2 * n) where
   p _  0     = 1
   p [] _     = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Mar 16 2012

Crossrefs

Cf. A035294, A079126, A000009, A079124, A079125, A067953, A111133.

Sequence in context: A227547 A163685 A080614 * A265069 A265070 A071679

Adjacent sequences: A079119 A079120 A079121 * A079123 A079124 A079125

Keyword

nonn

Author

Reinhard Zumkeller, Dec 27 2002

Status

approved