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Number of partitions of n with rank 2 (the rank of a partition is the largest part minus the number of parts).
4

%I #23 May 26 2023 11:12:44

%S 0,0,1,0,1,1,2,2,3,3,6,6,9,10,15,16,23,27,36,42,55,64,84,98,124,147,

%T 185,217,270,318,391,461,562,661,802,942,1132,1331,1592,1864,2220,

%U 2597,3077,3593,4240,4940,5811,6758,7916,9192,10737,12438,14488,16755,19459,22465,26024,29987

%N Number of partitions of n with rank 2 (the rank of a partition is the largest part minus the number of parts).

%C Column k=2 in the triangle A063995.

%D George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

%H Seiichi Manyama, <a href="/A101199/b101199.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(9/2) * n^(3/2)). - _Vaclav Kotesovec_, May 26 2023

%e a(6)=1 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111 have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively.

%p with(combinat): for n from 1 to 45 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=2 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n],n=1..45);

%t Table[Count[Max[#]-Length[#]&/@IntegerPartitions[n],2],{n,60}] (* _Harvey P. Dale_, Dec 22 2018 *)

%Y Cf. A000041, A063995, A101198, A101200.

%K nonn

%O 1,7

%A _Emeric Deutsch_, Dec 12 2004

%E More terms from _Joerg Arndt_, Oct 07 2012