A390431 Number of multisets summing to n that do not contain 1 and are not the first sums of any multiset.

0, 0, 0, 0, 0, 1, 0, 1, 1, 4, 3, 6, 7, 14, 15, 23, 28, 43, 51, 72, 88, 120, 146, 193, 236, 307, 373, 476, 578, 727, 879, 1092, 1318, 1623, 1949, 2380, 2846, 3452, 4115, 4955, 5888, 7053, 8350, 9952, 11744, 13931, 16389, 19361, 22711, 26730, 31265, 36666, 42773

Offset

1,10

Comments

The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

Links

Table of n, a(n) for n=1..53.

Example

The multiset {2,3,3} is not the first sums of any multiset, so it is counted under a(8). Note the first sums of (1,1,2,1) are (2,3,3), but (1,1,2,1) is not a multiset.
The a(6) = 1 through a(12) = 6 multisets:
  {3,3}  .  {2,3,3}  {3,3,3}  {5,5}      {2,4,5}    {2,5,5}
                              {2,4,4}    {3,3,5}    {3,3,6}
                              {3,3,4}    {2,3,3,3}  {2,2,4,4}
                              {2,2,3,3}             {2,3,3,4}
                                                    {3,3,3,3}
                                                    {2,2,2,3,3}

Mathematica

uncha[tar_, ini_]:=(cur={ini}; Do[AppendTo[cur, tar[[k]]-If[k==1, ini, cur[[k]]]], {k, 1, Length[tar]}]; cur);
greseq[m_]:=Select[Table[uncha[m, i], {i, 1, Max[m]}], LessEqual@@#&]!={};
Table[Length[Select[Reverse/@IntegerPartitions[n], FreeQ[#, 1]&&!greseq[#]&]], {n, 30}]

Crossrefs

These are multisets that do not contain 1 and are not a row of A390307.

The complement without 1 is counted by A390446, ranks A390448.

Allowing 1 gives A390447, ranked by A390445, complement A390449.

The partitions are ranked by A390569.

A000041 counts integer partitions, strict A000009.

A342527 counts compositions with all equal first sums.

A390429 counts distinct possible first sums of partitions of n.

A390567 counts compositions with all distinct first sums.

Cf. A049988, A325325, A368684, A389811, A389919, A390444.

Sequence in context: A103476 A021700 A309516 * A211362 A211246 A197829

Adjacent sequences: A390428 A390429 A390430 * A390432 A390433 A390434

Keyword

nonn

Author

Gus Wiseman, Nov 19 2025

Status

approved