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 A015723 Number of parts in all partitions of n into distinct parts. 42
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe) Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018. Arnold Knopfmacher, and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18. Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See s(n). Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol. FORMULA 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 G.f.: sum(k>=1, k*x^(k*(k+1)/2)/prod(i=1..k, 1-x^i ) ). - Vladeta Jovovic, Sep 21 2005 a(n) = A238131(n)+A238132(n) = sum_{k=1..n} A048272(k)*A000009(n-k). - Mircea Merca, Feb 26 2014 a(n) = Sum_{k>=1} k*A008289(n,k). - Vaclav Kotesovec, Apr 16 2016 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 For n > 0, a(n) = A116676(n) + A116680(n). - Vaclav Kotesovec, May 26 2018 EXAMPLE 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 MAPLE 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]: seq(a(n), n=1..50);  # Alois P. Heinz, Feb 27 2013 MATHEMATICA 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

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Last modified April 18 22:45 EDT 2021. Contains 343098 sequences. (Running on oeis4.)