A305552 Number of uniform normal multiset partitions of weight n.

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

Andrew Howroyd, Table of n, a(n) for n = 0..1000

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

Crossrefs

Cf. A000005, A001315, A007716, A034691, A038041, A074854, A289078, A305552, A306017.

Sequence in context: A126471 A317100 A199932 * A226652 A024696 A295360

Adjacent sequences: A305549 A305550 A305551 * A305553 A305554 A305555

Keyword

nonn

Author

Gus Wiseman, Jun 20 2018

Status

approved