A372542 Total number of modes in all partitions of n.

0, 1, 2, 4, 6, 9, 16, 20, 30, 42, 61, 76, 112, 138, 189, 248, 325, 407, 539, 667, 865, 1083, 1369, 1690, 2140, 2624, 3268, 4009, 4954, 6022, 7417, 8968, 10946, 13218, 16023, 19256, 23264, 27819, 33415, 39873, 47682, 56654, 67527, 79962, 94909, 112130, 132578

Offset: 0

Author

Alois P. Heinz, May 05 2024

Keywords

• nonn

Comments

Each element of a partition with maximal multiplicity is a mode of this partition.

Examples

			a(5) = 9 = 1 + 2 + 2 + 1 + 1 + 1 + 1: 5, 32, 41, 221, 311, 2111, 11111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000041, A003056, A362614, A372632.

Programs

Maple

b:= proc(n, i, m, t) option remember; `if`(n=0, t, `if`(i<1, 0, add(
      b(n-i*j, i-1, max(j, m), `if`(j>m, 1, `if`(j=m, t+1, t))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..46);

Formula

a(n) = Sum_{k=0..A003056(n)} k * A362614(n,k).