A360671 Number of multisets whose right half (inclusive) sums to n.

1, 2, 5, 8, 16, 21, 42, 51, 90, 121, 185, 235, 386, 465, 679, 908, 1261, 1580, 2238, 2770, 3827, 4831, 6314, 7910, 10619, 13074, 16813, 21049, 26934, 33072, 42445, 51679, 65264, 79902, 99309, 121548, 151325, 182697, 224873, 272625, 334536, 401999, 491560, 588723

Offset

0,2

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1 + Sum_{k>=1} x^k*(2 - x^k)/((1 - x^k)^(k+1) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023

Example

The a(0) = 1 through a(4) = 16 multisets:
  {}  {1}    {2}        {3}            {4}
      {1,1}  {1,2}      {1,3}          {1,4}
             {2,2}      {2,3}          {2,4}
             {1,1,1}    {3,3}          {3,4}
             {1,1,1,1}  {1,1,2}        {4,4}
                        {1,1,1,2}      {1,1,3}
                        {1,1,1,1,1}    {1,2,2}
                        {1,1,1,1,1,1}  {2,2,2}
                                       {1,1,1,3}
                                       {1,1,2,2}
                                       {1,2,2,2}
                                       {2,2,2,2}
                                       {1,1,1,1,2}
                                       {1,1,1,1,1,2}
                                       {1,1,1,1,1,1,1}
                                       {1,1,1,1,1,1,1,1}
For example, the multiset y = {1,1,1,1,2} has right half (inclusive) {1,1,2}, with sum 4, so y is counted under a(4).

Mathematica

Table[Length[Select[Join@@IntegerPartitions/@Range[0, 3*k], Total[Take[#, Ceiling[Length[#]/2]]]==k&]], {k, 0, 15}]

Prog

(PARI) seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k*(2-x^k)/(1-x^k + O(x*x^(n-k)))^(k+1); p /= 1 - x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023

Crossrefs

The exclusive version is A360673.

Column sums of A360675 with rows reversed.

The case of sets is A360955, exclusive A360954.

The even-length case is A360956.

A360672 counts partitions by left sum (exclusive).

A360679 gives right sum (inclusive) of prime indices.

Cf. A000041, A347044, A361200, A361201.

Sequence in context: A080084 A065093 A361270 * A168470 A295998 A129299

Adjacent sequences: A360668 A360669 A360670 * A360672 A360673 A360674

Keyword

nonn

Author

Gus Wiseman, Mar 09 2023

Extensions

Terms a(24) and beyond from Andrew Howroyd, Mar 11 2023

Status

approved