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

0,6

References

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

Links

Table of n, a(n) for n=0..55.

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.

Example

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).

Mathematica

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

Crossrefs

Cf. A000041, A101707, A064173.

Cf. A101198-A101200, A101709.

Sequence in context: A054082 A339381 A238544 * A339559 A248880 A026255

Adjacent sequences: A101705 A101706 A101707 * A101709 A101710 A101711

Keyword

nonn

Author

Emeric Deutsch, Dec 12 2004

Extensions

More terms from Joerg Arndt, Oct 07 2012

Offset changed to 0 by Georg Fischer, Dec 23 2023

Status

approved