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
Keywords
Examples
a(4)=1 [1+3+4=2*4]; a(5)=3 [1+2+3+4=1+4+5=2+3+5=2*5].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..80 from Reinhard Zumkeller)
Programs
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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
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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 *)
Formula
a(n) = b(0, n), b(m, n) = 1 + sum(b(i, j): m
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