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) = 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
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
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
Row lengths of A325537.
KEYWORD
nonn
AUTHOR
EXTENSIONS
Extended and corrected by Naohiro Nomoto, Feb 24 2002
STATUS
approved