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A232697 Number of partitions of 2n into parts such that the largest multiplicity equals n. 6
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Dec 18 2016 *)

CROSSREFS

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

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

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Last modified August 10 02:24 EDT 2020. Contains 336365 sequences. (Running on oeis4.)