A101199 Number of partitions of n with rank 2 (the rank of a partition is the largest part minus the number of parts).

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, 185, 217, 270, 318, 391, 461, 562, 661, 802, 942, 1132, 1331, 1592, 1864, 2220, 2597, 3077, 3593, 4240, 4940, 5811, 6758, 7916, 9192, 10737, 12438, 14488, 16755, 19459, 22465, 26024, 29987

Offset

1,7

Comments

Column k=2 in the triangle A063995.

References

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 26 2023

Example

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.

Maple

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);

Mathematica

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

Crossrefs

Cf. A000041, A063995, A101198, A101200.

Sequence in context: A035642 A213332 A133392 * A032155 A116932 A240579

Adjacent sequences: A101196 A101197 A101198 * A101200 A101201 A101202

Keyword

nonn

Author

Emeric Deutsch, Dec 12 2004

Extensions

More terms from Joerg Arndt, Oct 07 2012

Status

approved