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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<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

PROG

(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 ) );

Vec(gf) /* Joerg Arndt, Oct 20 2012 */

CROSSREFS

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

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

Row lengths of A325537.

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

Sequence in context: A105888 A123632 A039868 * A116645 A177739 A323581

Adjacent sequences:  A015720 A015721 A015722 * A015724 A015725 A015726

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Extended and corrected by Naohiro Nomoto, Feb 24 2002

STATUS

approved

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Last modified October 15 22:25 EDT 2019. Contains 328038 sequences. (Running on oeis4.)