OFFSET
1,2
COMMENTS
The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
FORMULA
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
EXAMPLE
The a(1) = 1 through a(8) = 10 multisets:
{1,1} {1,2} {1,3} {1,4} {1,5} {1,6} {1,7} {1,8}
{2,2} {2,3} {2,4} {2,5} {2,6} {2,7} {2,8}
{3,3} {3,4} {3,5} {3,6} {3,7} {3,8}
{1,1,1} {4,4} {4,5} {4,6} {4,7} {4,8}
{5,5} {5,6} {5,7} {5,8}
{1,1,2} {6,6} {6,7} {6,8}
{1,2,2} {7,7} {7,8}
{2,2,2} {1,1,3} {8,8}
{1,1,1,1} {1,2,3}
{2,2,3}
MATHEMATICA
zz[n_]:=Select[IntegerPartitions[n], UnsameQ@@#&&GreaterEqual @@ Differences[Append[#, 0]]&];
Table[Sum[Append[z, 0][[1]]-Append[z, 0][[2]], {z, zz[n]}], {n, 30}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2023
STATUS
approved