A218074 - OEIS

%I #33 Oct 27 2023 21:10:15

%S 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 333,399,475,568,673,794,938,1102,1289,1512,1760,2050,2384,2760,3190,

%U 3687,4246,4882,5609,6427,7354,8412,9592,10927,12439,14130,16033,18177,20573,23256,26271

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

%C 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 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 a(n) = A015723(n) - A000009(n) for n>0. - _Alois P. Heinz_, Sep 03 2014

%e a(8) = 7 because in the 6 partitions of 8 into distinct parts

%e   1:  [ 1 2 5 ]

%e   2:  [ 1 3 4 ]

%e   3:  [ 1 7 ]

%e   4:  [ 2 6 ]

%e   5:  [ 3 5 ]

%e   6:  [ 8 ]

%e there are 2+2+1+1+1+0 = 7 up-steps. - _Joerg Arndt_, Sep 03 2014

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

%p       b(n, i-1)+`if`(i>n, 0, (p->p+[0, p[1]])(b(n-i, i-1)))))

%p     end:

%p a:= n-> `if`(n=0, 0, (p-> p[2]-p[1])(b(n$2))):

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Sep 03 2014

%t 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 (PARI)

%o N=66;  q='q+O('q^N);

%o gf=sum(n=1,N, (n-1)*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );

%o v=Vec(gf+'a0);  v[1]-='a0;  v  /* include initial zeros */

%Y Cf. A015723, Sum_{n>=0} (n * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).

%Y Cf. A032020, Sum_{n>=0} (n! * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).

%Y Cf. A032153, Sum_{n>=1} ((n-1)! * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).

%Y Cf. A072576, Sum_{n>=0} ((n+1)! * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).

%Y Cf. A058884 (up-steps in all partitions).

%K nonn

%O 0,6

%A _Joerg Arndt_, Oct 20 2012