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A218074 Expansion of Sum_{n>=1} ((n-1) * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).

%I A218074 #33 Oct 27 2023 21:10:15
%S A218074 0,0,0,1,1,2,4,5,7,10,15,18,25,31,41,53,66,81,103,125,154,190,229,276,
%T A218074 333,399,475,568,673,794,938,1102,1289,1512,1760,2050,2384,2760,3190,
%U A218074 3687,4246,4882,5609,6427,7354,8412,9592,10927,12439,14130,16033,18177,20573,23256,26271
%N A218074 Expansion of Sum_{n>=1} ((n-1) * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
%C A218074 Number of up-steps (== number of parts - 1) in all partitions of n into distinct parts (represented as increasing lists), see example. - _Joerg Arndt_, Sep 03 2014
%H A218074 Alois P. Heinz, <a href="/A218074/b218074.txt">Table of n, a(n) for n = 0..2000</a> (first 201 terms from Vincenzo Librandi)
%F A218074 a(n) = A015723(n) - A000009(n) for n>0. - _Alois P. Heinz_, Sep 03 2014
%e A218074 a(8) = 7 because in the 6 partitions of 8 into distinct parts
%e A218074   1:  [ 1 2 5 ]
%e A218074   2:  [ 1 3 4 ]
%e A218074   3:  [ 1 7 ]
%e A218074   4:  [ 2 6 ]
%e A218074   5:  [ 3 5 ]
%e A218074   6:  [ 8 ]
%e A218074 there are 2+2+1+1+1+0 = 7 up-steps. - _Joerg Arndt_, Sep 03 2014
%p A218074 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A218074       b(n, i-1)+`if`(i>n, 0, (p->p+[0, p[1]])(b(n-i, i-1)))))
%p A218074     end:
%p A218074 a:= n-> `if`(n=0, 0, (p-> p[2]-p[1])(b(n$2))):
%p A218074 seq(a(n), n=0..80);  # _Alois P. Heinz_, Sep 03 2014
%t A218074 max=80; s=Sum[(n-1)*q^(n*(n+1)/2)/QPochhammer[q, q, n], {n, Sqrt[max+1]}]+ O[q]^max; CoefficientList[s, q] (* _Jean-François Alcover_, Jan 17 2016 *)
%o A218074 (PARI)
%o A218074 N=66;  q='q+O('q^N);
%o A218074 gf=sum(n=1,N, (n-1)*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
%o A218074 v=Vec(gf+'a0);  v[1]-='a0;  v  /* include initial zeros */
%Y A218074 Cf. A015723, Sum_{n>=0} (n * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
%Y A218074 Cf. A032020, Sum_{n>=0} (n! * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
%Y A218074 Cf. A032153, Sum_{n>=1} ((n-1)! * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
%Y A218074 Cf. A072576, Sum_{n>=0} ((n+1)! * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
%Y A218074 Cf. A058884 (up-steps in all partitions).
%K A218074 nonn
%O A218074 0,6
%A A218074 _Joerg Arndt_, Oct 20 2012