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A211881 Difference between sum of largest parts and sum of smallest parts of all partitions of n into an even number of parts.

%I A211881 #13 Feb 15 2017 07:55:19
%S A211881 0,0,0,1,2,5,9,16,26,41,65,95,142,202,293,403,568,766,1054,1399,1886,
%T A211881 2469,3276,4237,5538,7094,9162,11628,14856,18704,23670,29590,37130,
%U A211881 46109,57428,70885,87685,107634,132324,161595,197545,240091,291990,353302,427624
%N A211881 Difference between sum of largest parts and sum of smallest parts of all partitions of n into an even number of parts.
%H A211881 Alois P. Heinz, <a href="/A211881/b211881.txt">Table of n, a(n) for n = 0..1000</a>
%F A211881 a(n) = A222048(n) - A222045(n).
%F A211881 a(n) = A116686(n) - A211870(n).
%e A211881 a(6) = 9: 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], difference between sum of largest parts and sum of smallest parts is (1+2+3+3+4+5) - (1+1+1+3+2+1) = 18 - 9 = 9.
%p A211881 g:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
%p A211881       `if`(i>n, [0, 0], g(n, i+1)+(l-> [l[2], l[1]])(g(n-i, i)))
%p A211881     end:
%p A211881 b:= proc(n, i) option remember;
%p A211881       [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
%p A211881        `if`(n<i, [0, 0], (l-> [l[2], l[1]])(b(n-i, i))))
%p A211881     end:
%p A211881 a:= n-> g(n, 1)[2] -b(n, n)[2]:
%p A211881 seq(a(n), n=0..50);
%t A211881 g[n_, i_] := g[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, g[n, i+1] + Reverse[g[n-i, i]]]; b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[n<i, {0, 0}, Reverse[b[n-i, i]]]]; a[n_] := g[n, 1][[2]] - b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 15 2017, translated from Maple *)
%Y A211881 Cf. A116686, A211870, A222045, A222048.
%K A211881 nonn
%O A211881 0,5
%A A211881 _Alois P. Heinz_, Feb 13 2013