OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{d|n} Sum_{j=1..n/d} binomial(d + A008284(n/d, j) - 1, d) for n > 0. - Andrew Howroyd, Dec 31 2022
EXAMPLE
The a(1) = 1 through a(6) = 17 multiset partitions:
{1} {2} {3} {4} {5} {6}
{11} {12} {13} {14} {15}
{1}{1} {111} {22} {23} {24}
{1}{1}{1} {112} {113} {33}
{1111} {122} {114}
{2}{2} {1112} {123}
{11}{11} {11111} {222}
{1}{1}{1}{1} {1}{1}{1}{1}{1} {1113}
{1122}
{3}{3}
{11112}
{111111}
{12}{12}
{2}{2}{2}
{111}{111}
{11}{11}{11}
{1}{1}{1}{1}{1}{1}
MATHEMATICA
Table[If[n==0, 1, Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d], n/d], SameQ@@Length/@#&], {d, Divisors[n]}]]]], {n, 0, 20}]
PROG
(PARI)
P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
seq(n) = {my(u=Vec(P(n, y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ Andrew Howroyd, Dec 31 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 05 2022
EXTENSIONS
Terms a(41) and beyond from Andrew Howroyd, Dec 31 2022
STATUS
approved