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

Author

Gus Wiseman, Mar 09 2023

Keywords

• nonn

Examples

			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).
		

Links

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

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.

Programs

Mathematica

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

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

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

Extensions

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