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Number of overpartitions of n minus the number of partitions of n.
8

%I #16 Jul 28 2015 07:55:05

%S 0,1,2,5,9,17,29,49,78,124,190,288,427,627,905,1296,1831,2567,3563,

%T 4910,6709,9112,12286,16473,21953,29108,38388,50398,65850,85683,

%U 111020,143302,184263,236113,301498,383757,486909,615955,776921,977263,1225934,1533945

%N Number of overpartitions of n minus the number of partitions of n.

%C Number of overpartitions of n that contain at least one overlined part. - _Omar E. Pol_, Jan 19 2014

%H Alois P. Heinz, <a href="/A230441/b230441.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A015128(n) - A000041(n).

%e The 14 overpartitions of 4 are

%e 01: [4],

%e 02: [4'],

%e 03: [2, 2],

%e 04: [2', 2],

%e 05: [3, 1],

%e 06: [3', 1],

%e 07: [3, 1'],

%e 08: [3', 1'],

%e 09: [2, 1, 1],

%e 10: [2', 1, 1],

%e 11: [2, 1', 1],

%e 12: [2', 1', 1],

%e 13: [1, 1, 1, 1],

%e 14: [1', 1, 1, 1].

%e There are 9 overpartitions that contain at least one overlined part, so a(4) = 9. - _Omar E. Pol_, Jan 19 2014

%p b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0$2],

%p b(n, i-1) +add((l->l+[0, l[2]])(b(n-i*j, i-1)), j=1..n/i)))

%p end:

%p a:= n-> (l->l[2]-l[1])(b(n$2)):

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jan 30 2014

%t 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 Cf. A000041, A015128, A235792.

%K nonn

%O 0,3

%A _Omar E. Pol_, Jan 09 2014