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A359041
Number of finite sets of integer partitions with all equal sums and total sum n.
1
1, 1, 2, 3, 6, 7, 14, 15, 32, 31, 63, 56, 142, 101, 240, 211, 467, 297, 985, 490, 1524, 1247, 2542, 1255, 6371, 1979, 7486, 7070, 14128, 4565, 32953, 6842, 42229, 37863, 56266, 17887, 192914, 21637, 145820, 197835, 371853, 44583, 772740, 63261, 943966, 1124840
OFFSET
0,3
FORMULA
a(n) = Sum_{d|n} binomial(A000041(d),n/d).
EXAMPLE
The a(1) = 1 through a(6) = 14 sets:
{(1)} {(2)} {(3)} {(4)} {(5)} {(6)}
{(11)} {(21)} {(22)} {(32)} {(33)}
{(111)} {(31)} {(41)} {(42)}
{(211)} {(221)} {(51)}
{(1111)} {(311)} {(222)}
{(2),(11)} {(2111)} {(321)}
{(11111)} {(411)}
{(2211)}
{(3111)}
{(21111)}
{(111111)}
{(3),(21)}
{(3),(111)}
{(21),(111)}
MATHEMATICA
Table[If[n==0, 1, Sum[Binomial[PartitionsP[d], n/d], {d, Divisors[n]}]], {n, 0, 50}]
PROG
(PARI) a(n) = if (n, sumdiv(n, d, binomial(numbpart(d), n/d)), 1); \\ Michel Marcus, Dec 14 2022
CROSSREFS
This is the constant-sum case of A261049, ordered A358906.
The version for all different sums is A271619, ordered A336342.
Allowing repetition gives A305551, ordered A279787.
The version for compositions instead of partitions is A358904.
A001970 counts multisets of partitions.
A034691 counts multisets of compositions, ordered A133494.
A098407 counts sets of compositions, ordered A358907.
Sequence in context: A066880 A075427 A075426 * A191615 A018606 A117087
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 14 2022
STATUS
approved