A066189 Sum of all partitions of n into distinct parts.

0, 1, 2, 6, 8, 15, 24, 35, 48, 72, 100, 132, 180, 234, 308, 405, 512, 646, 828, 1026, 1280, 1596, 1958, 2392, 2928, 3550, 4290, 5184, 6216, 7424, 8880, 10540, 12480, 14784, 17408, 20475, 24048, 28120, 32832, 38298, 44520, 51660, 59892, 69230, 79904

Offset: 0

Author

Wouter Meeussen, Dec 15 2001

Keywords

• easy
• nonn

Examples

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with sum 6+5+1+4+2+3+2+1 = 24. - _Gus Wiseman_, May 09 2019
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Row sums of A026793, A118457, A246688, A325537.

Cf. A015723, A022629, A066186, A147655, A325504, A325505, A325506, A325513, A325515, A325537.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i>n, [0$2],
      b(n, i+1)+(p-> p+[0, i*p[1]])(b(n-i, i+1))))
    end:
a:= n-> b(n, 1)[2]:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 01 2014

Mathematica

PartitionsQ[ Range[ 60 ] ]Range[ 60 ]
nmax=60; CoefficientList[Series[x*D[Product[1+x^k, {k, 1, nmax}], x], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 21 2016 *)

Formula

G.f.: sum(n>=1, n*q^(n-1)/(1+q^n) ) * prod(n>=1, 1+q^n ). - Joerg Arndt, Aug 03 2011

a(n) = n * A000009(n). - Vaclav Kotesovec, Sep 25 2016

G.f.: x*f'(x), where f(x) = Product_{k>=1} (1 + x^k). - Vaclav Kotesovec, Nov 21 2016

a(n) = A056239(A325506(n)). - Gus Wiseman, May 09 2019