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

0,3

Links

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

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

Example

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).

Mathematica

Table[Length[Select[Join@@IntegerPartitions/@Range[0, 3*k], EvenQ[Length[#]]&&Total[Take[#, 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/(1-x^k + O(x*x^(n-k)))^(k+1); p /= 1 - x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023

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.

Sequence in context: A308759 A137395 A001767 * A273160 A285138 A310018

Adjacent sequences: A360953 A360954 A360955 * A360957 A360958 A360959

Keyword

nonn

Author

Gus Wiseman, Mar 09 2023

Extensions

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

Status

approved