A204217 G.f.: Sum_{n>=1} n * x^(n*(n+1)/2) / (1 - x^n).

1, 1, 3, 1, 3, 4, 3, 1, 6, 5, 3, 4, 3, 5, 11, 1, 3, 8, 3, 6, 12, 5, 3, 4, 8, 5, 12, 8, 3, 13, 3, 1, 12, 5, 15, 12, 3, 5, 12, 6, 3, 15, 3, 9, 26, 5, 3, 4, 10, 10, 12, 9, 3, 17, 18, 8, 12, 5, 3, 17, 3, 5, 28, 1, 18, 19, 3, 9, 12, 17, 3, 13, 3, 5, 27, 9, 21, 20, 3, 6, 21, 5, 3, 19, 18, 5, 12, 12, 3

Offset

1,3

Comments

Conjecture: a(n) is the total number of parts in all partitions of n into consecutive parts. - Omar E. Pol, Apr 23 2017

Conjecture: row sums of A285914. - Omar E. Pol, Apr 30 2017

(The conjectures are true. See Joerg Arndt's proof in the Links section.) - Omar E. Pol, Jun 14 2017

Row lengths of A299765. - Omar E. Pol, Jul 23 2018

From Omar E. Pol, Oct 20 2019: (Start)

Row sums of A328361.

a(n) = 3 iff n is an odd prime. (End)

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (First 1024 terms from Paul D. Hanna)

Joerg Arndt, Proof of the conjectures of A204217, SeqFan Mailing List, Jun 03 2017.

Michael H. Mertens, Ken Ono and Larry Rolen, Mock modular Eisenstein series with Nebentypus, arXiv:1906.07410 [math.NT], 2019, p. 4, 9.

Formula

a(k) = 1 iff k = 2^n for n>=0.

G.f.: (1/Theta4(x)^2) * Sum_{n>=1} (-1)^(n-1)* n*x^(n*(n+1)/2) * (1 - x^n)/(1 + x^n)^2 where Theta4(x) = 1 + 2*Sum_{n>=1} (-x)^(n^2), due to an identity of Ramanujan.

Conjecture: a(n) = Sum_{k=1..n} k*A285898(n,k). - R. J. Mathar, Apr 30 2017

Conjecture: a(n) = Sum_{k=1..A003056(n)} k*A237048(n,k). - Omar E. Pol, Apr 30 2017

(The conjectures are true. See Joerg Arndt's proof in the Links section.) - Omar E. Pol, Jun 14 2017

Example

G.f.: A(x) = x + x^2 + 3*x^3 + x^4 + 3*x^5 + 4*x^6 + 3*x^7 + x^8 + ...
follows by expanding A(x) = x/(1-x) + 2*x^3/(1-x^2) + 3*x^6/(1-x^3) + 4*x^10/(1-x^4) + ...
Also, by a Ramanujan identity:
A(x)*Theta4(x)^2 = x*(1-x)/(1+x)^2 - 2*x^3*(1-x^2)/(1+x^2)^2 + 3*x^6*(1-x^3)/(1+x^3)^2 - 4*x^10*(1-x^4)/(1+x^4)^2 + 5*x^15*(1-x^5)/(1+x^5)^2 + ...
For n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. The total number of parts is 11, so a(15) = 11. - Omar E. Pol, Apr 23 2017
From Omar E. Pol, Nov 30 2020: (Start)
Illustration of initial terms:
                         Diagram
n   a(n)                       _
1     1                      _|1
2     1                    _|1 _
3     3                  _|1  |2
4     1                _|1   _|
5     3              _|1    |2 _
6     4            _|1     _| |3
7     3          _|1      |2  |
8     1        _|1       _|  _|
9     6      _|1        |2  |3 _
10    5     |1          |   | |4
...
a(n) is the total length of all vertical line segments that are below and that share one vertex with the horizontal line segments that are in the n-th level of the diagram. For more information about the diagram see A286001 and A237593. (End)

Mathematica

terms = 1024; Sum[n*x^(n*(n+1)/2)/(1-x^n), {n, 1, Ceiling[Sqrt[2*terms]]}] + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Jun 04 2017 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=1, n, m*x^(m*(m+1)/2)/(1-x^m+x*O(x^n))), n)}
(PARI) {a(n)=local(Theta4=1+2*sum(m=1, sqrtint(n+1), (-x)^(m^2))+x*O(x^n)); polcoeff(1/Theta4^2*sum(m=1, sqrtint(2*n+1), (-1)^(m-1)*m*x^(m*(m+1)/2)*(1-x^m)/(1+x^m+x*O(x^n))^2), n)}
(PARI) a(n) = {nb = 0; forpart(v=n, nbp = #v; if ((#Set(v)==#v) && (v[nbp] - v[1] == #v-1), nb += #v); ); nb; } \\ Michel Marcus, Apr 23 2017
(PARI) a(n) = {my(i=2, t=1); n--; while(n>0, t += (i*(n%i==0)); n-=i; i++); t} \\ David A. Corneth, Apr 28 2017

Crossrefs

Cf. A204218, A237593, A285914, A286001, A299765, A328361, A357618.

Sequence in context: A337936 A296388 A081772 * A296955 A219525 A050121

Adjacent sequences: A204214 A204215 A204216 * A204218 A204219 A204220

Keyword

nonn,easy

Author

Paul D. Hanna, Jan 12 2012

Status

approved