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

%I A360673 #15 Mar 11 2023 15:07:42
%S A360673 1,2,7,13,27,37,73,89,156,205,315,387,644,749,1104,1442,2015,2453,
%T A360673 3529,4239,5926,7360,9624,11842,16115,19445,25084,31137,39911,48374,
%U A360673 62559,75135,95263,115763,143749,174874,218614,261419,321991,388712,477439,569968,698493
%N A360673 Number of multisets of positive integers whose right half (exclusive) sums to n.
%H A360673 Andrew Howroyd, <a href="/A360673/b360673.txt">Table of n, a(n) for n = 0..1000</a>
%F A360673 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
%e A360673 The a(0) = 1 through a(3) = 13 multisets:
%e A360673   {}  {1,1}    {1,2}        {1,3}
%e A360673       {1,1,1}  {2,2}        {2,3}
%e A360673                {1,1,2}      {3,3}
%e A360673                {1,2,2}      {1,1,3}
%e A360673                {2,2,2}      {1,2,3}
%e A360673                {1,1,1,1}    {1,3,3}
%e A360673                {1,1,1,1,1}  {2,2,3}
%e A360673                             {2,3,3}
%e A360673                             {3,3,3}
%e A360673                             {1,1,1,2}
%e A360673                             {1,1,1,1,2}
%e A360673                             {1,1,1,1,1,1}
%e A360673                             {1,1,1,1,1,1,1}
%e A360673 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).
%t A360673 Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], Total[Take[#,Floor[Length[#]/2]]]==k&]],{k,0,15}]
%o A360673 (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
%Y A360673 The inclusive version is A360671.
%Y A360673 Column sums of A360672.
%Y A360673 The case of sets is A360954, inclusive A360955.
%Y A360673 The even-length case is A360956.
%Y A360673 A359893 and A359901 count partitions by median.
%Y A360673 First for prime indices, second for partitions, third for prime factors:
%Y A360673 - A360676 gives left sum (exclusive), counted by A360672, product A361200.
%Y A360673 - A360677 gives right sum (exclusive), counted by A360675, product A361201.
%Y A360673 - A360678 gives left sum (inclusive), counted by A360675, product A347043.
%Y A360673 - A360679 gives right sum (inclusive), counted by A360672, product A347044.
%Y A360673 Cf. A000041, A360616, A360617, A360674, A360675, A360953.
%K A360673 nonn
%O A360673 0,2
%A A360673 _Gus Wiseman_, Mar 04 2023
%E A360673 Terms a(21) and beyond from _Andrew Howroyd_, Mar 11 2023