A388064 - OEIS

A388064

Total number of parts in all partitions of n with designated summands.

0, 1, 5, 12, 29, 54, 111, 189, 344, 563, 946, 1483, 2382, 3611, 5567, 8256, 12335, 17891, 26117, 37187, 53161, 74572, 104738, 144878, 200557, 273995, 374314, 505937, 683192, 914245, 1222094, 1620684, 2146336, 2823348, 3707698, 4840427, 6308514, 8178558, 10584331

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OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..5000

EXAMPLE

For n = 4 there are 29 parts in all partitions of 4 with designated summands. They are [4'], [3'+ 1'], [2'+ 2], [2 + 2'], [2'+ 1'+ 1], [2'+ 1 + 1'], [1'+ 1 + 1 + 1], [1 + 1'+ 1 + 1], [1 + 1 + 1'+ 1], [1 + 1 + 1 + 1'], so a(4) = 29.

MAPLE

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

b(n, i-1)+add((p-> p+[0, p[1]*j])(b(n-i*j, i-1)*j), j=1..n/i)))

end:

a:= n-> b(n$2)[2]:

seq(a(n), n=0..38); # Alois P. Heinz, Oct 09 2025

CROSSREFS

Row sums of A388063.