A390447 Number of multisets summing to n not equal to the first sums of any nonempty multiset.

1, 1, 2, 3, 5, 8, 11, 16, 23, 34, 45, 62, 84, 115, 150, 199, 259, 340, 436, 562, 715, 912, 1148, 1448, 1811, 2265, 2809, 3486, 4296, 5292, 6483, 7934, 9667, 11766, 14259, 17263, 20823, 25089, 30130, 36140, 43226, 51636, 61524, 73213, 86919, 103065, 121947

Offset

1,3

Comments

Here, a multiset is a finite weakly increasing sequence of positive integers.

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..47.

Example

The a(1) = 1 through a(8) = 16 multisets:
  {1}  {11}  {12}   {13}    {14}     {15}      {16}       {17}
             {111}  {112}   {113}    {33}      {115}      {116}
                    {1111}  {122}    {114}     {124}      {125}
                            {1112}   {123}     {133}      {134}
                            {11111}  {1113}    {1114}     {233}
                                     {1122}    {1123}     {1115}
                                     {11112}   {1222}     {1124}
                                     {111111}  {11113}    {1133}
                                               {11122}    {1223}
                                               {111112}   {11114}
                                               {1111111}  {11123}
                                                          {11222}
                                                          {111113}
                                                          {111122}
                                                          {1111112}
                                                          {11111111}

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], greseq]], {n, 30}]

Crossrefs

These are multisets that are not a row of A390307.

The case without 1 is A390431, ranks A390569.

These partitions are ranked by A390445, complement A390449.

The complement is counted by A390446, ranks A390448.

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, A325546, A368684, A389811, A389919, A390362, A390444.

Sequence in context: A344650 A238591 A039847 * A046938 A060677 A131787

Adjacent sequences: A390444 A390445 A390446 * A390448 A390449 A390450

Keyword

nonn

Author

Gus Wiseman, Nov 13 2025

Status

approved