A359678 - OEIS

%I #10 Jan 22 2023 11:33:55

%S 1,2,4,4,6,9,8,10,14,13,16,21,17,22,28,23,30,37,30,38,46,38,46,59,46,

%T 55,70,59,70,86,67,81,96,84,98,115,95,114,135,114,132,158,127,156,178,

%U 148,176,207,172,201,227,196,228,270,222,255,296,255,295,338,278

%N Number of multisets (finite weakly increasing sequences of positive integers) with zero-based weighted sum (A359674) equal to n > 0.

%C The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

%H Andrew Howroyd, <a href="/A359678/b359678.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum_{k>=2} x^binomial(k,2)/((1 - x^binomial(k,2))*Product_{j=1..k-1} (1 - x^(binomial(k,2)-binomial(j,2)))). - _Andrew Howroyd_, Jan 22 2023

%e The a(1) = 1 through a(8) = 10 multisets:

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

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

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

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

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

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

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

%e                                          {2,2,2}    {1,1,3}  {8,8}

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

%e                                                              {2,2,3}

%t zz[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&&GreaterEqual @@ Differences[Append[#,0]]&];

%t Table[Sum[Append[z,0][[1]]-Append[z,0][[2]],{z,zz[n]}],{n,30}]

%o (PARI) seq(n)={Vec(sum(k=2, (sqrtint(8*n+1)+1)\2, my(t=binomial(k, 2)); x^t/((1-x^t)*prod(j=1, k-1, 1 - x^(t-binomial(j, 2)) + O(x^(n-t+1))))))} \\ _Andrew Howroyd_, Jan 22 2023

%Y The one-based version is A320387.

%Y Number of appearances of n > 0 in A359674.

%Y The sorted minimal ranks are A359675, reverse A359680.

%Y The minimal ranks are A359676, reverse A359681.

%Y The maximal ranks are A359757.

%Y A053632 counts compositions by zero-based weighted sum.

%Y A124757 gives zero-based weighted sums of standard compositions, rev A231204.

%Y Cf. A029931, A243055, A304818, A358194, A359677.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 15 2023