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A222049 Difference between sums of largest parts of all partitions of n into odd number of parts and into even number of parts.

%I A222049 #17 Apr 13 2018 19:32:51
%S A222049 0,1,1,2,0,2,-1,2,-4,4,-6,5,-9,8,-12,14,-19,19,-22,26,-32,38,-41,48,
%T A222049 -56,65,-70,84,-95,107,-115,133,-153,172,-186,212,-240,264,-289,325,
%U A222049 -366,400,-437,485,-544,597,-649,714,-799,869,-942,1037,-1148,1246,-1351
%N A222049 Difference between sums of largest parts of all partitions of n into odd number of parts and into even number of parts.
%H A222049 Alois P. Heinz, <a href="/A222049/b222049.txt">Table of n, a(n) for n = 0..6000</a>
%F A222049 a(n) = A222047(n) - A222048(n).
%F A222049 G.f.: Sum_{i>=0} i*x^i/Product_{j=1..i} (1 + x^j). - _Ilya Gutkovskiy_, Apr 13 2018
%e A222049 a(6) = -1 = (2+2+3+4+6) - (1+2+3+3+4+5) because the partitions of 6 into an odd number of parts are [2,1,1,1,1], [2,2,2], [3,2,1], [4,1,1], [6] and the partitions of 6 into an even number of parts are [1,1,1,1,1,1], [2,2,1,1], [3,1,1,1], [3,3], [4,2], [5,1].
%p A222049 b:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
%p A222049       `if`(i>n, [0, 0], b(n, i+1)+(l-> [l[2], l[1]])(b(n-i, i)))
%p A222049     end:
%p A222049 a:= n-> (l->l[1]-l[2])(b(n, 1)):
%p A222049 seq(a(n), n=0..60);
%t A222049 b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, b[n, i+1] + Reverse @ b[n-i, i]]; a[n_] :=  b[n, 1][[1]]-b[n, 1][[2]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 02 2017, translated from Maple *)
%Y A222049 Cf. A211373, A222044, A222045, A222046, A222047, A222048.
%K A222049 sign
%O A222049 0,4
%A A222049 _Alois P. Heinz_, Feb 06 2013