A015723 - OEIS

%I #80 May 21 2021 04:17:15

%S 1,1,3,3,5,8,10,13,18,25,30,40,49,63,80,98,119,149,179,218,266,318,

%T 380,455,541,640,760,895,1050,1234,1442,1679,1960,2272,2635,3052,3520,

%U 4054,4669,5359,6142,7035,8037,9170,10460,11896,13517,15349,17394,19691

%N Number of parts in all partitions of n into distinct parts.

%H Alois P. Heinz, <a href="/A015723/b015723.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.

%H Arnold Knopfmacher, and Neville Robbins, <a href="http://www.plouffe.fr/OEIS/citations/robbinspart.pdf">Identities for the total number of parts in partitions of integers</a>, Util. Math. 67 (2005), 9-18.

%H Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See s(n).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PolygammaFunction.html">q-Polygamma Function</a>, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.

%F G.f.: sum(k>=1, x^k/(1+x^k) ) * prod(m>=1, 1+x^m ). Convolution of A048272 and A000009. - _Vladeta Jovovic_, Nov 26 2002

%F G.f.: sum(k>=1, k*x^(k*(k+1)/2)/prod(i=1..k, 1-x^i ) ). - _Vladeta Jovovic_, Sep 21 2005

%F a(n) = A238131(n)+A238132(n) = sum_{k=1..n} A048272(k)*A000009(n-k). - _Mircea Merca_, Feb 26 2014

%F a(n) = Sum_{k>=1} k*A008289(n,k). - _Vaclav Kotesovec_, Apr 16 2016

%F 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

%F a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (2 * Pi * n^(1/4)). - _Vaclav Kotesovec_, May 19 2018

%F For n > 0, a(n) = A116676(n) + A116680(n). - _Vaclav Kotesovec_, May 26 2018

%e 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

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

%p       add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 1))))

%p     end:

%p a:= n-> b(n, n)[2]:

%p seq(a(n), n=1..50);  # _Alois P. Heinz_, Feb 27 2013

%t 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 *)

%t 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 *)

%t Table[Length[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,30}] - _Gus Wiseman_, May 09 2019

%t b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0},

%t    Sum[{#[[1]], #[[2]] + #[[1]]*j}&@ b[n-i*j, i-1], {j, 0, Min[n/i, 1]}]]];

%t a[n_] := b[n, n][[2]];

%t Array[a, 50] (* _Jean-François Alcover_, May 21 2021, after _Alois P. Heinz_ *)

%o (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 ) );

%o Vec(gf) /* _Joerg Arndt_, Oct 20 2012 */

%Y Cf. A006128, A008289, A079499, A067619, A186545.

%Y Column k=1 of: A210485, A213177, A327622.

%Y Row lengths of A325537.

%Y Cf. A022629, A066186, A066189, A325504, A325505, A325506, A325513, A325515.

%K nonn

%O 1,3

%A _Clark Kimberling_

%E Extended and corrected by _Naohiro Nomoto_, Feb 24 2002