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A015723
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Number of parts in all partitions of n into distinct parts.
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77
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1, 1, 3, 3, 5, 8, 10, 13, 18, 25, 30, 40, 49, 63, 80, 98, 119, 149, 179, 218, 266, 318, 380, 455, 541, 640, 760, 895, 1050, 1234, 1442, 1679, 1960, 2272, 2635, 3052, 3520, 4054, 4669, 5359, 6142, 7035, 8037, 9170, 10460, 11896, 13517, 15349, 17394, 19691
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f.: sum(k>=1, k*x^(k*(k+1)/2)/prod(i=1..k, 1-x^i ) ). - Vladeta Jovovic, Sep 21 2005
G.f.: -(-1; x)_inf * (log(1-x) + psi_x(1 - log(-1)/log(x)))/(2*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - Vladimir Reshetnikov, Nov 21 2016
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (2 * Pi * n^(1/4)). - Vaclav Kotesovec, May 19 2018
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EXAMPLE
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The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with a total of 1 + 2 + 2 + 3 = 8 parts, so a(6) = 8. - Gus Wiseman, May 09 2019
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 1))))
end:
a:= n-> b(n, n)[2]:
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MATHEMATICA
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nn=50; Rest[CoefficientList[Series[D[Product[1+y x^i, {i, 1, nn}], y]/.y->1, {x, 0, nn}], x]] (* Geoffrey Critzer, Oct 29 2012; fixed by Vaclav Kotesovec, Apr 16 2016 *)
q[n_, k_] := q[n, k] = If[n<k || k<1, 0, If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]]; Table[Sum[k*q[n, k], {k, 1, Floor[(Sqrt[8*n+1] - 1)/2]}], {n, 1, 100}] (* Vaclav Kotesovec, Apr 16 2016 *)
Table[Length[Join@@Select[IntegerPartitions[n], UnsameQ@@#&]], {n, 0, 30}] - Gus Wiseman, May 09 2019
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0},
Sum[{#[[1]], #[[2]] + #[[1]]*j}&@ b[n-i*j, i-1], {j, 0, Min[n/i, 1]}]]];
a[n_] := b[n, n][[2]];
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PROG
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(PARI) N=66; q='q+O('q^N); gf=sum(n=0, N, n*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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