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A218074
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Expansion of Sum_{n>=1} ((n-1) * q^(n*(n+1)/2) / Product_{k=1..n} (1 - q^k)).
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4
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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
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OFFSET
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0,6
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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
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MAPLE
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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))):
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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