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Number of partitions of n having positive even rank (the rank of a partition is the largest part minus the number of parts).
17

%I #18 Dec 23 2023 13:49:28

%S 0,0,0,1,0,2,1,4,3,7,6,14,13,23,24,41,43,67,75,111,126,177,204,282,

%T 328,437,514,674,793,1021,1207,1533,1814,2273,2691,3344,3956,4865,

%U 5754,7027,8296,10060,11864,14302,16836,20183,23715,28301,33191,39423,46152,54607,63794,75200,87687,103018

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

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

%F 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

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

%e 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).

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

%Y Cf. A000041, A101707, A064173.

%Y Cf. A101198-A101200, A101709.

%K nonn

%O 0,6

%A _Emeric Deutsch_, Dec 12 2004

%E More terms from _Joerg Arndt_, Oct 07 2012

%E Offset changed to 0 by _Georg Fischer_, Dec 23 2023