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

%I A232697 #24 Jun 26 2022 16:55:52
%S A232697 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 A232697 321,384,479,575,709,848,1040,1239,1508,1795,2168,2574,3095,3661,4379,
%U A232697 5171,6154,7246,8592,10088,11915,13960,16425,19197,22520,26253,30702,35718
%N A232697 Number of partitions of 2n into parts such that the largest multiplicity equals n.
%H A232697 Alois P. Heinz, <a href="/A232697/b232697.txt">Table of n, a(n) for n = 0..1000</a>
%F A232697 G.f.: x/(1-x) + Product_{k>=2} 1/(1-x^k).
%F A232697 a(0) = 1, a(n) = 1 + A002865(n) = 1 + A000041(n)-A000041(n-1) for n>0.
%F A232697 a(n) = A091602(2n,n) = A096144(2n,n).
%F A232697 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). - _Vaclav Kotesovec_, Oct 25 2018
%e A232697 a(1) = 1: [2].
%e A232697 a(2) = 2: [2,2], [2,1,1].
%e A232697 a(3) = 2: [2,2,2], [3,1,1,1].
%e A232697 a(4) = 3: [2,2,2,2], [2,2,1,1,1,1], [4,1,1,1,1].
%e A232697 a(5) = 3: [2,2,2,2,2], [3,2,1,1,1,1,1], [5,1,1,1,1,1].
%e A232697 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 A232697 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A232697       `if`(i>n, 0, add(b(n-i*j, i+1, min(k,
%p A232697        iquo(n-i*j, i+1))), j=0..min(n/i, k))))
%p A232697     end:
%p A232697 a:= n-> b(2*n, 1, n)-`if`(n=0, 0, b(2*n, 1, n-1)):
%p A232697 seq(a(n), n=0..60);
%t A232697 CoefficientList[x/(1-x) + (1-x)/QPochhammer[x] + O[x]^60, x] (* _Jean-François Alcover_, Dec 18 2016 *)
%Y A232697 Cf. A000041, A002865, A091602, A232623, A332051.
%Y A232697 Partial sums give A133041.
%K A232697 nonn
%O A232697 0,3
%A A232697 _Alois P. Heinz_, Nov 27 2013