login
A305552
Number of uniform normal multiset partitions of weight n.
3
1, 1, 3, 5, 12, 17, 47, 65, 170, 277, 655, 1025, 2739, 4097, 10281, 17257, 41364, 65537, 170047, 262145, 660296, 1094457, 2621965, 4194305, 10898799, 16792721, 41945103, 69938141, 168546184, 268435457, 694029255, 1073741825, 2696094037, 4474449261, 10737451027
OFFSET
0,3
COMMENTS
A multiset is normal if it spans an initial interval of positive integers. A multiset partition m is uniform if all parts have the same size, and normal if all parts are normal. The weight of m is the sum of sizes of its parts.
LINKS
FORMULA
a(n) = Sum_{d|n} binomial(2^(n/d - 1) + d - 1, d).
EXAMPLE
The a(4) = 12 uniform normal multiset partitions:
{1111}, {1222}, {1122}, {1112}, {1233}, {1223}, {1123}, {1234},
{11,11}, {11,12}, {12,12},
{1,1,1,1}.
MATHEMATICA
Table[Sum[Binomial[2^(n/k-1)+k-1, k], {k, Divisors[n]}], {n, 35}]
PROG
(PARI) a(n)={if(n<1, n==0, sumdiv(n, d, binomial(2^(n/d - 1) + d - 1, d)))} \\ Andrew Howroyd, Jun 22 2018
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 20 2018
STATUS
approved