A372888 Sum of binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

0, 1, 2, 7, 13, 31, 66, 138, 279, 581, 1173, 2375, 4783, 9630, 19316, 38802, 77689, 155673, 311639, 623845, 1248179, 2497719, 4996387, 9995304, 19992908, 39990902, 79986136, 159983241, 319975073, 639971495, 1279962115, 2559966847, 5119970499, 10240030209

Offset: 0

Author

Gus Wiseman, May 23 2024

Keywords

• nonn

Examples

			The strict partitions of 6 are (6), (5,1), (4,2), (3,2,1), with respective binary ranks 32, 17, 10, 7 with sum 66, so a(6) = 66.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3321

Crossrefs

Row sums of A118462 (binary ranks of strict partitions).

For Heinz number the non-strict version is A145519, row sums of A215366.

For Heinz number (not binary rank) we have A147655, row sums of A246867.

The non-strict version is A372890.

A000009 counts strict partitions, ranks A005117.

A048675 gives binary rank of prime indices, distinct A087207.

A277905 groups all positive integers by binary rank of prime indices.

Binary indices (A048793):

- length A000120, complement A023416

- min A001511, opposite A000012

- max A029837 or A070939, opposite A070940

- sum A029931, product A096111

- reverse A272020

- complement A368494, sum A359400

- opposite A371572, sum A230877

- opposite complement A371571, sum A359359

Cf. A000041, A005940, A015716, A018819, A019565, A118457, A225620, A344086.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2 [0, p[1]*2^(i-1)]
          +p)(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..33);  # Alois P. Heinz, May 23 2024

Mathematica

Table[Total[Total[2^(#-1)]& /@ Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]

Formula

a(n) = Sum_{k=1..n} 2^(k-1) * A015716(n,k). - Alois P. Heinz, May 24 2024