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A304797 Expansion of x * (d/dx) Sum_{k>=0} k!*x^(k*(k+1)/2)/Product_{j=1..k} (1 - x^j).

%I A304797 #6 May 18 2018 19:49:06
%S A304797 0,1,2,9,12,25,66,91,152,243,570,715,1212,1729,2702,5265,6960,10489,
%T A304797 15318,22363,31100,57771,72534,109411,151032,219025,293930,421281,
%U A304797 680820,883369,1256010,1727971,2396000,3235419,4447506,5894875,9266580,11691001,16380470,21774753
%N A304797 Expansion of x * (d/dx) Sum_{k>=0} k!*x^(k*(k+1)/2)/Product_{j=1..k} (1 - x^j).
%C A304797 Sum of all parts of all compositions (ordered partitions) of n into distinct parts.
%H A304797 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A304797 a(n) = n*A032020(n).
%p A304797 b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
%p A304797      `if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
%p A304797     end:
%p A304797 a:= n-> n*add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
%p A304797 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 18 2018
%t A304797 nmax = 39; CoefficientList[Series[x D[Sum[k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], x], {x, 0, nmax}], x]
%Y A304797 Cf. A001787, A032020, A066186, A066189, A097910, A303664.
%K A304797 nonn
%O A304797 0,3
%A A304797 _Ilya Gutkovskiy_, May 18 2018