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A239292 (sum of all odd parts of all strict partitions of n) - (sum of all even parts of all strict partitions of n); for "strict", see Comments.

%I A239292 #14 May 31 2021 05:32:46
%S A239292 0,1,-2,2,0,3,-4,-1,4,4,0,0,4,-2,0,1,16,6,4,2,8,8,14,4,20,18,22,32,32,
%T A239292 32,28,32,52,56,64,83,76,92,112,130,140,168,172,198,212,256,288,318,
%U A239292 368,416,456,527,564,640,712,806,884,985,1116,1224,1344,1496
%N A239292 (sum of all odd parts of all strict partitions of n) - (sum of all even parts of all strict partitions of n); for "strict", see Comments.
%C A239292 A strict partition is one having distinct parts.  a(n) < 0 if and only if n is one of these:  2,6,7,13.
%H A239292 Alois P. Heinz, <a href="/A239292/b239292.txt">Table of n, a(n) for n = 0..1000</a>
%F A239292 a(n) = A116682(n) - A116684(n) for n >= 0.
%e A239292 The strict partitions of 6 are 6, 51, 42, 321.  The sum of all the odd parts is 10 and the sum of all the even parts is 14, so that a(6) = -4.
%p A239292 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
%p A239292       `if`(n=0, [1, 0], b(n, i-1) +`if`(i>n, 0, (p->p+
%p A239292       [0, p[1]*`if`(irem(i, 2)=1, i, -i)])(b(n-i, i-1)))))
%p A239292     end:
%p A239292 a:= n-> b(n$2)[2]:
%p A239292 seq(a(n), n=0..80);  # _Alois P. Heinz_, Mar 15 2014
%t A239292 d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; Map[Total[Select[#, OddQ]] - Total[Select[#, EvenQ]]&[Flatten[d[#]]] &, -1 + Range[55]]  (* _Peter J. C. Moses_, Mar 14 2014 *)
%t A239292 b[n_, i_] := b[n, i] = If[n > i (i + 1)/2, 0,
%t A239292      If[n == 0, {1, 0}, b[n, i - 1] + If[i > n, 0, Function[p, p +
%t A239292      {0, p[[1]]*If[Mod[i, 2] == 1, i, -i]}][b[n - i, i - 1]]]]];
%t A239292 a[n_] := b[n, n][[2]];
%t A239292 a /@ Range[0, 80] (* _Jean-François Alcover_, May 31 2021, after _Alois P. Heinz_ *)
%Y A239292 Cf. A000009, A116682, A116684.
%K A239292 sign,easy
%O A239292 0,3
%A A239292 _Clark Kimberling_, Mar 14 2014