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Number of partitions of 2n into parts such that the largest multiplicity equals n.
7

%I #24 Jun 26 2022 16:55:52

%S 1,1,2,2,3,3,5,5,8,9,13,15,22,25,35,42,56,67,89,106,138,166,211,254,

%T 321,384,479,575,709,848,1040,1239,1508,1795,2168,2574,3095,3661,4379,

%U 5171,6154,7246,8592,10088,11915,13960,16425,19197,22520,26253,30702,35718

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

%H Alois P. Heinz, <a href="/A232697/b232697.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). - _Vaclav Kotesovec_, Oct 25 2018

%e a(1) = 1: [2].

%e a(2) = 2: [2,2], [2,1,1].

%e a(3) = 2: [2,2,2], [3,1,1,1].

%e a(4) = 3: [2,2,2,2], [2,2,1,1,1,1], [4,1,1,1,1].

%e a(5) = 3: [2,2,2,2,2], [3,2,1,1,1,1,1], [5,1,1,1,1,1].

%e 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].

%p b:= proc(n, i, k) option remember; `if`(n=0, 1,

%p `if`(i>n, 0, add(b(n-i*j, i+1, min(k,

%p iquo(n-i*j, i+1))), j=0..min(n/i, k))))

%p end:

%p a:= n-> b(2*n, 1, n)-`if`(n=0, 0, b(2*n, 1, n-1)):

%p seq(a(n), n=0..60);

%t CoefficientList[x/(1-x) + (1-x)/QPochhammer[x] + O[x]^60, x] (* _Jean-François Alcover_, Dec 18 2016 *)

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

%Y Partial sums give A133041.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 27 2013