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%I #16 Jul 06 2019 17:29:43
%S 0,1,1,3,1,4,2,5,0,7,1,7,-1,8,-1,12,-5,11,-3,13,-8,18,-8,18,-14,22,
%T -14,28,-21,29,-20,34,-33,43,-33,49,-44,54,-48,67,-64,73,-66,85,-87,
%U 105,-94,114,-120,132,-128,156,-159,174,-172,203,-213,234,-232,263
%N Difference between sums of smallest parts of all partitions of n into odd number of parts and into even number of parts.
%H Vaclav Kotesovec, <a href="/A222046/b222046.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F a(n) = A222044(n) - A222045(n).
%F a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 06 2019
%e a(6) = 2 = (1+2+1+1+6) - (1+1+1+3+2+1) 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 b:= proc(n, i) option remember;
%p [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
%p `if`(n<i, [0, 0], (l-> [l[2], l[1]])(b(n-i, i))))
%p end:
%p a:= n-> (l->l[1]-l[2])(b(n, n)):
%p seq(a(n), n=0..100);
%t 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_] := (#[[1]] - #[[2]]&)[b[n, n] ]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jan 23 2017, translated from Maple *)
%Y Cf. A211373, A222044, A222045, A222047, A222048, A222049.
%K sign
%O 0,4
%A _Alois P. Heinz_, Feb 06 2013