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A230441 Number of overpartitions of n minus the number of partitions of n.

%I A230441 #16 Jul 28 2015 07:55:05
%S A230441 0,1,2,5,9,17,29,49,78,124,190,288,427,627,905,1296,1831,2567,3563,
%T A230441 4910,6709,9112,12286,16473,21953,29108,38388,50398,65850,85683,
%U A230441 111020,143302,184263,236113,301498,383757,486909,615955,776921,977263,1225934,1533945
%N A230441 Number of overpartitions of n minus the number of partitions of n.
%C A230441 Number of overpartitions of n that contain at least one overlined part. - _Omar E. Pol_, Jan 19 2014
%H A230441 Alois P. Heinz, <a href="/A230441/b230441.txt">Table of n, a(n) for n = 0..1000</a>
%F A230441 a(n) = A015128(n) - A000041(n).
%e A230441 The 14 overpartitions of 4 are
%e A230441 01: [4],
%e A230441 02: [4'],
%e A230441 03: [2, 2],
%e A230441 04: [2', 2],
%e A230441 05: [3, 1],
%e A230441 06: [3', 1],
%e A230441 07: [3, 1'],
%e A230441 08: [3', 1'],
%e A230441 09: [2, 1, 1],
%e A230441 10: [2', 1, 1],
%e A230441 11: [2, 1', 1],
%e A230441 12: [2', 1', 1],
%e A230441 13: [1, 1, 1, 1],
%e A230441 14: [1', 1, 1, 1].
%e A230441 There are 9 overpartitions that contain at least one overlined part, so a(4) = 9. - _Omar E. Pol_, Jan 19 2014
%p A230441 b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0$2],
%p A230441       b(n, i-1) +add((l->l+[0, l[2]])(b(n-i*j, i-1)), j=1..n/i)))
%p A230441     end:
%p A230441 a:= n-> (l->l[2]-l[1])(b(n$2)):
%p A230441 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jan 30 2014
%t A230441 b[n_, i_] := b[n, i] = If[n==0, {1, 1}, If[i<1, {0, 0}, b[n, i-1] + Sum[Function[ {l}, l+{0, l[[2]]}][b[n-i*j, i-1]], {j, 1, n/i}]]]; a[n_] := Function[{l}, l[[2]]-l[[1]]][b[n, n]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jul 28 2015, after _Alois P. Heinz_ *)
%Y A230441 Cf. A000041, A015128, A235792.
%K A230441 nonn
%O A230441 0,3
%A A230441 _Omar E. Pol_, Jan 09 2014