A232697 Number of partitions of 2n into parts such that the largest multiplicity equals n.

1, 1, 2, 2, 3, 3, 5, 5, 8, 9, 13, 15, 22, 25, 35, 42, 56, 67, 89, 106, 138, 166, 211, 254, 321, 384, 479, 575, 709, 848, 1040, 1239, 1508, 1795, 2168, 2574, 3095, 3661, 4379, 5171, 6154, 7246, 8592, 10088, 11915, 13960, 16425, 19197, 22520, 26253, 30702, 35718

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

G.f.: x/(1-x) + Product_{k>=2} 1/(1-x^k).

a(0) = 1, a(n) = 1 + A002865(n) = 1 + A000041(n)-A000041(n-1) for n>0.

a(n) = A091602(2n,n) = A096144(2n,n).

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). - Vaclav Kotesovec, Oct 25 2018

Example

a(1) = 1: [2].
a(2) = 2: [2,2], [2,1,1].
a(3) = 2: [2,2,2], [3,1,1,1].
a(4) = 3: [2,2,2,2], [2,2,1,1,1,1], [4,1,1,1,1].
a(5) = 3: [2,2,2,2,2], [3,2,1,1,1,1,1], [5,1,1,1,1,1].
a(6) = 5: [2,2,2,2,2,2], [2,2,2,1,1,1,1,1,1], [3,3,1,1,1,1,1,1], [4,2,1,1,1,1,1,1], [6,1,1,1,1,1,1].

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)-`if`(n=0, 0, b(2*n, 1, n-1)):
seq(a(n), n=0..60);

Mathematica

CoefficientList[x/(1-x) + (1-x)/QPochhammer[x] + O[x]^60, x] (* Jean-François Alcover, Dec 18 2016 *)

Crossrefs

Cf. A000041, A002865, A091602, A232623, A332051.

Partial sums give A133041.

Sequence in context: A239949 A103609 A237800 * A129526 A246998 A000358

Adjacent sequences: A232694 A232695 A232696 * A232698 A232699 A232700

Keyword

nonn

Author

Alois P. Heinz, Nov 27 2013

Status

approved