A360673 Number of multisets of positive integers whose right half (exclusive) sums to n.

1, 2, 7, 13, 27, 37, 73, 89, 156, 205, 315, 387, 644, 749, 1104, 1442, 2015, 2453, 3529, 4239, 5926, 7360, 9624, 11842, 16115, 19445, 25084, 31137, 39911, 48374, 62559, 75135, 95263, 115763, 143749, 174874, 218614, 261419, 321991, 388712, 477439, 569968, 698493

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+2) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023

Example

The a(0) = 1 through a(3) = 13 multisets:
  {}  {1,1}    {1,2}        {1,3}
      {1,1,1}  {2,2}        {2,3}
               {1,1,2}      {3,3}
               {1,2,2}      {1,1,3}
               {2,2,2}      {1,2,3}
               {1,1,1,1}    {1,3,3}
               {1,1,1,1,1}  {2,2,3}
                            {2,3,3}
                            {3,3,3}
                            {1,1,1,2}
                            {1,1,1,1,2}
                            {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 (exclusive) {1,2}, with sum 3, so y is counted under a(3).

Mathematica

Table[Length[Select[Join@@IntegerPartitions/@Range[0, 3*k], Total[Take[#, Floor[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+2); p /= 1 - x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023

Crossrefs

The inclusive version is A360671.

Column sums of A360672.

The case of sets is A360954, inclusive A360955.

The even-length case is A360956.

A359893 and A359901 count partitions by median.

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, A360616, A360617, A360674, A360675, A360953.

Sequence in context: A180470 A182415 A275491 * A183435 A360385 A141777

Adjacent sequences: A360670 A360671 A360672 * A360674 A360675 A360676

Keyword

nonn

Author

Gus Wiseman, Mar 04 2023

Extensions

Terms a(21) and beyond from Andrew Howroyd, Mar 11 2023

Status

approved