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 A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts. 18
 1, 0, 1, 1, 3, 3, 7, 8, 15, 18, 30, 37, 58, 71, 105, 131, 186, 230, 318, 393, 530, 653, 863, 1060, 1380, 1686, 2164, 2637, 3345, 4057, 5096, 6158, 7665, 9228, 11395, 13671, 16765, 20040, 24418, 29098, 35251, 41869, 50460, 59755, 71669, 84626, 101050 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also the number of integer partitions of n that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons. - Gus Wiseman, Oct 30 2018 LINKS FORMULA a(n) = A000041(n) - Sum_{i=0..floor((n-1)/2)} A000041(i) = A000041(n) - A000070(floor((n-1)/2)) = A110619(n, 2). a(2*n) = A209816(n). - Gus Wiseman, Oct 30 2018 EXAMPLE a(5) = 3 since 5 can be partitioned as 1+1+1+1+1, 2+1+1+1, or 2+2+1; not counted are 5, 4+1, or 3+2. a(6) = 7 since 6 can be partitioned as 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 2+2+2, 1+1+1+3, 1+2+3, 3+3; not counted are 1+1+4, 2+4, 1+5, 6. From Gus Wiseman, Oct 30 2018: (Start) The a(2) = 1 through a(8) = 15 partitions with no part larger than n/2:   (11)  (111)  (22)    (221)    (33)      (322)      (44)                (211)   (2111)   (222)     (331)      (332)                (1111)  (11111)  (321)     (2221)     (422)                                 (2211)    (3211)     (431)                                 (3111)    (22111)    (2222)                                 (21111)   (31111)    (3221)                                 (111111)  (211111)   (3311)                                           (1111111)  (4211)                                                      (22211)                                                      (32111)                                                      (41111)                                                      (221111)                                                      (311111)                                                      (2111111)                                                      (11111111) The a(2) = 1 through a(8) = 15 partitions into n/2 or fewer parts:   (2)  (3)  (4)   (5)   (6)    (7)    (8)             (22)  (32)  (33)   (43)   (44)             (31)  (41)  (42)   (52)   (53)                         (51)   (61)   (62)                         (222)  (322)  (71)                         (321)  (331)  (332)                         (411)  (421)  (422)                                (511)  (431)                                       (521)                                       (611)                                       (2222)                                       (3221)                                       (3311)                                       (4211)                                       (5111) The a(6) = 7 integer partitions of 6 with no part larger than n/2 together with a realizing set multipartition of each (the parts of the partition count the appearances of each vertex in the set multipartition):       (33): {{1,2},{1,2},{1,2}}      (321): {{1,2},{1,2},{1,3}}     (3111): {{1,2},{1,3},{1,4}}      (222): {{1,2,3},{1,2,3}}     (2211): {{1,2},{1,2,3,4}}    (21111): {{1,2},{1,3,4,5}}   (111111): {{1,2,3,4,5,6}} (End) MAPLE A000070 := proc(n) add( combinat[numbpart](i), i=0..n) ; end proc: A110618 := proc(n) combinat[numbpart](n) - A000070(floor((n-1)/2)) ; end proc: # R. J. Mathar, Jan 24 2011 MATHEMATICA f[n_, 1] := 1; f[1, k_] := 1; f[n_, k_] := f[n, k] = If[k > n, f[n, k - 1], f[n, k - 1] + f[n - k, k]]; g[n_] := f[n, Floor[n/2]]; g[0] = 1; g[1] = 0; Array[g, 47, 0] (* Robert G. Wilson v, Jan 23 2011 *) sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]]; multhyp[m_]:=Select[mps[m], And[And@@UnsameQ@@@#, Min@@Length/@#>1]&]; strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n]; Table[Length[Select[strnorm[n], multhyp[#]!={}&]], {n, 8}] (* Gus Wiseman, Oct 30 2018 *) PROG (PARI) a(n) = numbpart(n) - sum(i=0, if (n%2, n\2, n/2-1), numbpart(i)); \\ Michel Marcus, Oct 31 2018 CROSSREFS Cf. A000070, A000569, A025065, A049311, A096373, A116540, A147878, A209816, A283877, A306005, A320921. Sequence in context: A241642 A086543 A281616 * A320291 A320294 A304179 Adjacent sequences:  A110615 A110616 A110617 * A110619 A110620 A110621 KEYWORD nonn AUTHOR Henry Bottomley, Aug 01 2005 STATUS approved

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Last modified June 21 19:59 EDT 2021. Contains 345365 sequences. (Running on oeis4.)