A360955 - OEIS

%I #10 Mar 13 2023 17:47:26

%S 1,1,2,3,4,6,7,11,12,19,20,31,33,49,51,77,79,112,124,165,177,247,260,

%T 340,388,480,533,693,747,925,1078,1271,1429,1772,1966,2331,2705,3123,

%U 3573,4245,4737,5504,6424,7254,8256,9634,10889,12372,14251,16031,18379

%N Number of finite sets of positive integers whose right half (inclusive) sums to n.

%H Andrew Howroyd, <a href="/A360955/b360955.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{w>=1} Sum_{h=w..floor((n-binomial(w,2))/w)} binomial(h,w) * A072233(n - w*h - binomial(w,2), w-1) for n > 0. - _Andrew Howroyd_, Mar 13 2023

%e The a(1) = 1 through a(8) = 12 sets:

%e   {1}  {2}    {3}    {4}    {5}      {6}      {7}        {8}

%e        {1,2}  {1,3}  {1,4}  {1,5}    {1,6}    {1,7}      {1,8}

%e               {2,3}  {2,4}  {2,5}    {2,6}    {2,7}      {2,8}

%e                      {3,4}  {3,5}    {3,6}    {3,7}      {3,8}

%e                             {4,5}    {4,6}    {4,7}      {4,8}

%e                             {1,2,3}  {5,6}    {5,7}      {5,8}

%e                                      {1,2,4}  {6,7}      {6,8}

%e                                               {1,2,5}    {7,8}

%e                                               {1,3,4}    {1,2,6}

%e                                               {2,3,4}    {1,3,5}

%e                                               {1,2,3,4}  {2,3,5}

%e                                                          {1,2,3,5}

%e For example, the set y = {2,3,5} has right half (inclusive) {3,5}, with sum 8, so y is counted under a(8).

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

%o (PARI) \\ P(n,k) is A072233(n,k).

%o P(n,k)=polcoef(1/prod(k=1, k, 1 - x^k + O(x*x^n)), n)

%o a(n)=if(n==0, 1, sum(w=1, sqrt(n), my(t=binomial(w,2)); sum(h=w, (n-t)\w, binomial(h, w) * P(n-w*h-t, w-1)))) \\ _Andrew Howroyd_, Mar 13 2023

%Y The version for multisets is A360671, exclusive A360673.

%Y The exclusive version is A360954.

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

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

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

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

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

%Y Cf. A000009, A072233, A359893, A359901, A360674, A360956.

%K nonn

%O 0,3

%A _Gus Wiseman_, Mar 09 2023

%E Terms a(16) and beyond from _Andrew Howroyd_, Mar 13 2023