A101708 Number of partitions of n having positive even rank (the rank of a partition is the largest part minus the number of parts).

0, 0, 0, 1, 0, 2, 1, 4, 3, 7, 6, 14, 13, 23, 24, 41, 43, 67, 75, 111, 126, 177, 204, 282, 328, 437, 514, 674, 793, 1021, 1207, 1533, 1814, 2273, 2691, 3344, 3956, 4865, 5754, 7027, 8296, 10060, 11864, 14302, 16836, 20183, 23715, 28301, 33191, 39423, 46152, 54607, 63794, 75200, 87687, 103018

Offset: 0

Author

Emeric Deutsch, Dec 12 2004

Keywords

• nonn

Examples

			a(7)=4 because the only partitions of 7 with positive even rank are 7 (rank=6), 61 (rank=4), 511 (rank=2) and 43 (rank=2).
		

References

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

Crossrefs

Cf. A000041, A101707, A064173.

Cf. A101198-A101200, A101709.

Programs

Mathematica

Table[Count[Max[#]-Length[#]&/@IntegerPartitions[n],?(EvenQ[#] && Positive[#]&)], {n,50}] (* _Harvey P. Dale, Feb 26 2012 *)

Formula

G.f.: Sum((-1)^(k+1)*x^((3*k^2+3*k)/2)/(1+x^k), k>=1)/Product(1-x^k, k>=1). - Vladeta Jovovic, Dec 20 2004

a(n) = A064173(n) - A101707(n) for n >= 1.

Extensions

More terms from Joerg Arndt, Oct 07 2012

Offset changed to 0 by Georg Fischer, Dec 23 2023