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A236634 Number of unbalanced partitions of n: the largest part is not equal to the number of parts.

%I A236634 #16 Jan 11 2020 06:37:11
%S A236634 0,2,2,4,6,10,12,20,26,38,50,70,90,124,160,212,272,356,450,582,732,
%T A236634 932,1166,1470,1824,2280,2814,3486,4280,5268,6428,7864,9552,11614,
%U A236634 14044,16990,20450,24626,29524,35392,42272,50472,60060,71444,84734,100432,118736
%N A236634 Number of unbalanced partitions of n: the largest part is not equal to the number of parts.
%C A236634 Number of partitions of n whose rank is not 0.
%F A236634 a(n) = A000041(n) - A047993(n) = 2*A064173(n).
%e A236634 For n = 5 we have:
%e A236634 -------------------------------------------------------
%e A236634 Partitions    Largest    Number    Dyson's
%e A236634 of 5           part     of parts    rank       Type
%e A236634 -------------------------------------------------------
%e A236634 5                5    -    1    =    4      unbalanced
%e A236634 4+1              4    -    2    =    2      unbalanced
%e A236634 3+2              3    -    2    =    1      unbalanced
%e A236634 3+1+1            3    -    3    =    0      balanced
%e A236634 2+2+1            2    -    3    =   -1      unbalanced
%e A236634 2+1+1+1          2    -    4    =   -2      unbalanced
%e A236634 1+1+1+1+1        1    -    5    =   -4      unbalanced
%e A236634 -------------------------------------------------------
%e A236634 There are 6 partitions whose rank is not 0, so a(5) = 6.
%t A236634 P = PartitionsP;
%t A236634 a[n_] := P[n] - Sum[-(-1)^k (P[n - (3k^2 - k)/2] - P[n - (3k^2 + k)/2]), {k, 1, Floor[(1 + Sqrt[1 + 24n])/6]}];
%t A236634 a /@ Range[46] (* _Jean-François Alcover_, Jan 11 2020, after _Wouter Meeussen_ in A047993 *)
%Y A236634 Cf. A000041, A047993, A064173, A209616.
%K A236634 nonn
%O A236634 1,2
%A A236634 _Omar E. Pol_, Feb 18 2014