A232623 Number of partitions of 2*n into parts with multiplicity <= n.

1, 1, 4, 9, 19, 37, 70, 124, 216, 363, 597, 960, 1519, 2359, 3617, 5469, 8173, 12079, 17680, 25630, 36848, 52547, 74383, 104556, 146018, 202651, 279631, 383719, 523813, 711502, 961902, 1294552, 1734788, 2315171, 3077592, 4075658, 5377900, 7071523, 9267454

Offset: 0

Author

Alois P. Heinz, Nov 27 2013

Keywords

• nonn

Examples

			a(1) = 1: [2].
a(2) = 4: [2,1,1], [2,2], [3,1], [4].
a(3) = 9: [2,2,1,1], [2,2,2], [3,1,1,1], [3,2,1], [3,3], [4,1,1], [4,2], [5,1], [6].
a(4) = 19: [2,2,1,1,1,1], [2,2,2,1,1], [2,2,2,2], [3,2,1,1,1], [3,2,2,1], [3,3,1,1], [3,3,2], [4,1,1,1,1], [4,2,1,1], [4,2,2], [4,3,1], [4,4], [5,1,1,1], [5,2,1], [5,3], [6,1,1], [6,2], [7,1], [8].
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Cf. A000041, A061199, A232605, A232697.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+1, min(k,
       iquo(n-i*j, i+1))), j=0..min(n/i, k))))
    end:
a:= n-> b(2*n, 1, n):
seq(a(n), n=0..50);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i>n, 0, Sum[b[n-i*j, i+1, Min[k, Quotient[n-i*j, i+1]]], {j, 0, Min[n/i, k]}]]]; a[n_] := b[2*n, 1, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)
A232623[n_] := PartitionsP[2*n] - PartitionsP[n - 1];
Array[A232623, 50, 0] (* Paolo Xausa, Jul 17 2025 *)

PARI

a(n) = numbpart(2*n) - numbpart(n-1); \\ Michel Marcus, Jul 13 2025

Formula

a(n) = A061199(n,2*n).

a(n) ~ exp(2*Pi*sqrt(n/3))/(8*n*sqrt(3)). - Vaclav Kotesovec, Nov 27 2013

a(n) = A000041(2*n) - A000041(n-1). - Alan Michael Gómez Calderón, Jul 12 2025