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

0,3

Links

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

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

Example

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

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 *)

Crossrefs

Row sums of A026793, A118457, A246688, A325537.

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

Sequence in context: A128913 A093005 A049818 * A278834 A336896 A306906

Adjacent sequences: A066186 A066187 A066188 * A066190 A066191 A066192

Keyword

easy,nonn

Author

Wouter Meeussen, Dec 15 2001

Status

approved