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

0, 0, 0, 1, 1, 2, 4, 5, 7, 10, 15, 18, 25, 31, 41, 53, 66, 81, 103, 125, 154, 190, 229, 276, 333, 399, 475, 568, 673, 794, 938, 1102, 1289, 1512, 1760, 2050, 2384, 2760, 3190, 3687, 4246, 4882, 5609, 6427, 7354, 8412, 9592, 10927, 12439, 14130, 16033, 18177, 20573, 23256, 26271

Offset: 0

Author

Joerg Arndt, Oct 20 2012

Keywords

• nonn

Comments

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

Examples

			a(8) = 7 because in the 6 partitions of 8 into distinct parts
  1:  [ 1 2 5 ]
  2:  [ 1 3 4 ]
  3:  [ 1 7 ]
  4:  [ 2 6 ]
  5:  [ 3 5 ]
  6:  [ 8 ]
there are 2+2+1+1+1+0 = 7 up-steps. - _Joerg Arndt_, Sep 03 2014
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 201 terms from Vincenzo Librandi)

Crossrefs

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

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

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

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

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

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, (p->p+[0, p[1]])(b(n-i, i-1)))))
    end:
a:= n-> `if`(n=0, 0, (p-> p[2]-p[1])(b(n$2))):
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 03 2014

Mathematica

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 *)

PARI

N=66;  q='q+O('q^N);
gf=sum(n=1,N, (n-1)*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
v=Vec(gf+'a0);  v[1]-='a0;  v  /* include initial zeros */

Formula

a(n) = A015723(n) - A000009(n) for n>0. - Alois P. Heinz, Sep 03 2014