A360956 Number of finite even-length multisets of positive integers whose right half sums to n.

1, 1, 3, 5, 10, 13, 26, 31, 55, 73, 112, 140, 233, 276, 405, 539, 750, 931, 1327, 1627, 2259, 2839, 3708, 4624, 6237, 7636, 9823, 12275, 15715, 19227, 24735, 30000, 37930, 46339, 57574, 70374, 87704, 105606, 129998, 157417, 193240, 231769, 283585, 339052, 411682, 493260

Offset: 0

Author

Gus Wiseman, Mar 09 2023

Keywords

• nonn

Examples

			The a(1) = 1 through a(5) = 13 multisets:
  {1,1}  {1,2}      {1,3}          {1,4}              {1,5}
         {2,2}      {2,3}          {2,4}              {2,5}
         {1,1,1,1}  {3,3}          {3,4}              {3,5}
                    {1,1,1,2}      {4,4}              {4,5}
                    {1,1,1,1,1,1}  {1,1,1,3}          {5,5}
                                   {1,1,2,2}          {1,1,1,4}
                                   {1,2,2,2}          {1,1,2,3}
                                   {2,2,2,2}          {1,2,2,3}
                                   {1,1,1,1,1,2}      {2,2,2,3}
                                   {1,1,1,1,1,1,1,1}  {1,1,1,1,1,3}
                                                      {1,1,1,1,2,2}
                                                      {1,1,1,1,1,1,1,2}
                                                      {1,1,1,1,1,1,1,1,1,1}
For example, the multiset y = {1,2,2,3} has right half {2,3}, with sum 5, so y is counted under a(5).
		

Links

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

Crossrefs

This is the even-length case of A360671 and A360673.

First for prime indices, second for partitions, third for prime factors:

- A360676 gives left sum (exclusive), counted by A360672, product A361200.

- A360677 gives right sum (exclusive), counted by A360675, product A361201.

- A360678 gives left sum (inclusive), counted by A360675, product A347043.

- A360679 gives right sum (inclusive), counted by A360672, product A347044.

Cf. A000041, A360674, A360954, A360955.

Programs

Mathematica

Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], EvenQ[Length[#]]&&Total[Take[#,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/(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/((1 - x^k)^(k+1) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023

Extensions

Terms a(16) and beyond from Andrew Howroyd, Mar 11 2023