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A359678
Number of multisets (finite weakly increasing sequences of positive integers) with zero-based weighted sum (A359674) equal to n > 0.
27
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, 55, 70, 59, 70, 86, 67, 81, 96, 84, 98, 115, 95, 114, 135, 114, 132, 158, 127, 156, 178, 148, 176, 207, 172, 201, 227, 196, 228, 270, 222, 255, 296, 255, 295, 338, 278
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
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
The one-based version is A320387.
Number of appearances of n > 0 in A359674.
The sorted minimal ranks are A359675, reverse A359680.
The minimal ranks are A359676, reverse A359681.
The maximal ranks are A359757.
A053632 counts compositions by zero-based weighted sum.
A124757 gives zero-based weighted sums of standard compositions, rev A231204.
Sequence in context: A241064 A292671 A210948 * A008133 A237828 A362607
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2023
STATUS
approved