cp's OEIS Frontend

A101200 Number of partitions of n with rank 3 (the rank of a partition is the largest part minus the number of parts).

%I A101200 #14 Feb 11 2025 18:22:48
%S A101200 0,0,0,1,0,1,1,2,2,4,3,6,7,10,11,17,18,26,30,40,47,63,72,94,111,140,
%T A101200 165,209,244,304,359,440,519,634,743,901,1060,1273,1494,1789,2092,
%U A101200 2491,2914,3449,4026,4752,5530,6502,7561,8852,10272,11997,13889,16171,18695,21700,25041,29002
%N A101200 Number of partitions of n with rank 3 (the rank of a partition is the largest part minus the number of parts).
%C A101200 Column k=3 in the triangle A063995.
%D A101200 George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
%H A101200 Seiichi Manyama, <a href="/A101200/b101200.txt">Table of n, a(n) for n = 1..1000</a>
%e A101200 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 A101200 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])=3 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n],n=1..45);
%t A101200 Table[Count[IntegerPartitions[n],_?(#[[1]]-Length[#]==3&)],{n,60}] (* _Harvey P. Dale_, Feb 11 2025 *)
%Y A101200 Cf. A000041, A063995, A101198, A101199.
%K A101200 nonn
%O A101200 1,8
%A A101200 _Emeric Deutsch_, Dec 12 2004
%E A101200 More terms, _Joerg Arndt_, Oct 07 2012