A390446 Number of multisets summing to n that are the first sums of some nonempty multiset.

0, 1, 1, 2, 2, 3, 4, 6, 7, 8, 11, 15, 17, 20, 26, 32, 38, 45, 54, 65, 77, 90, 107, 127, 147, 171, 201, 232, 269, 312, 359, 415, 476, 544, 624, 714, 814, 926, 1055, 1198, 1357, 1538, 1737, 1962, 2215, 2493, 2807, 3158, 3541, 3971, 4452, 4980, 5569, 6222, 6942, 7742

Offset

1,4

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..56.

Example

The multiset {1,3,3,5} has first sums {4,6,8}, so {4,6,8} is counted under a(18). We could also have used {2,2,4,4}.
The a(2) = 1 through a(9) = 7 multisets:
  {2}  {3}  {4}    {5}    {6}      {7}      {8}        {9}
            {2,2}  {2,3}  {2,4}    {2,5}    {2,6}      {2,7}
                          {2,2,2}  {3,4}    {3,5}      {3,6}
                                   {2,2,3}  {4,4}      {4,5}
                                            {2,2,4}    {2,2,5}
                                            {2,2,2,2}  {2,3,4}
                                                       {2,2,2,3}

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

Prog

(PARI) f(s, x, y) = sum(i=2*x, s\2, f(s-i, max(i-y, (i+1)\2), i-x)) + (s>=2*x);
a(n) = f(n, 1, n); \\ Jinyuan Wang, Nov 28 2025

Crossrefs

For partitions we have A368684.

The complement not containing 1 is counted by A390431.

The complement is counted by A390447, ranks A390445.

These multisets are ranked by A390448.

A000041 counts integer partitions, strict A000009.

A390307 gives first sums of prime indices.

A390429 counts distinct possible first sums of partitions.

Cf. A011782, A049988, A325325, A342527, A389811, A390567.

Sequence in context: A145813 A240865 A039859 * A130291 A067835 A029011

Adjacent sequences: A390443 A390444 A390445 * A390447 A390448 A390449

Keyword

nonn

Author

Gus Wiseman, Nov 13 2025

Extensions

More terms from Jinyuan Wang, Nov 28 2025

Status

approved