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A066189 Sum of all partitions of n into distinct parts.

%I A066189 #29 May 09 2019 15:02:38
%S A066189 0,1,2,6,8,15,24,35,48,72,100,132,180,234,308,405,512,646,828,1026,
%T A066189 1280,1596,1958,2392,2928,3550,4290,5184,6216,7424,8880,10540,12480,
%U A066189 14784,17408,20475,24048,28120,32832,38298,44520,51660,59892,69230,79904
%N A066189 Sum of all partitions of n into distinct parts.
%H A066189 Alois P. Heinz, <a href="/A066189/b066189.txt">Table of n, a(n) for n = 0..10000</a>
%F A066189 G.f.: sum(n>=1, n*q^(n-1)/(1+q^n) ) * prod(n>=1, 1+q^n ). - _Joerg Arndt_, Aug 03 2011
%F A066189 a(n) = n * A000009(n). - _Vaclav Kotesovec_, Sep 25 2016
%F A066189 G.f.: x*f'(x), where f(x) = Product_{k>=1} (1 + x^k). - _Vaclav Kotesovec_, Nov 21 2016
%F A066189 a(n) = A056239(A325506(n)). - _Gus Wiseman_, May 09 2019
%e A066189 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
%p A066189 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i>n, [0$2],
%p A066189       b(n, i+1)+(p-> p+[0, i*p[1]])(b(n-i, i+1))))
%p A066189     end:
%p A066189 a:= n-> b(n, 1)[2]:
%p A066189 seq(a(n), n=0..80);  # _Alois P. Heinz_, Sep 01 2014
%t A066189 PartitionsQ[ Range[ 60 ] ]Range[ 60 ]
%t A066189 nmax=60; CoefficientList[Series[x*D[Product[1+x^k, {k, 1, nmax}], x], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 21 2016 *)
%Y A066189 Row sums of A026793, A118457, A246688, A325537.
%Y A066189 Cf. A015723, A022629, A066186, A147655, A325504, A325505, A325506, A325513, A325515, A325537.
%K A066189 easy,nonn
%O A066189 0,3
%A A066189 _Wouter Meeussen_, Dec 15 2001