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A015723
Number of parts in all partitions of n into distinct parts.
77
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
OFFSET
1,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
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, 1, 50}] (* 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]];
Array[a, 50] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)
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
KEYWORD
nonn
EXTENSIONS
Extended and corrected by Naohiro Nomoto, Feb 24 2002
STATUS
approved