A026905 Partial sums of the partition numbers A000041 of the positive integers.

1, 3, 6, 11, 18, 29, 44, 66, 96, 138, 194, 271, 372, 507, 683, 914, 1211, 1596, 2086, 2713, 3505, 4507, 5762, 7337, 9295, 11731, 14741, 18459, 23024, 28628, 35470, 43819, 53962, 66272, 81155, 99132, 120769, 146784, 177969, 215307, 259890, 313064, 376325, 451500

Offset

1,2

Comments

Equivalently, a(n) = number of sums S of positive integers satisfying S <= n.

Equivalently, first differences give A000041. - Jacques ALARDET, Aug 04 2008, Aug 15 2008

For the partial sums of the partitions numbers of nonnegative integers A001477 see A000070. - Omar E. Pol, Nov 12 2011

Also number of parts in all regions of n that contain 1 as a part (Cf. A206437). - Omar E. Pol, Mar 11 2012

Also the number of graph minors of the path graph P_n (not counting the null graph). - Eric W. Weisstein, Apr 29 2022

Links

Table of n, a(n) for n=1..44.

Riccardo Aragona, Roberto Civino, and Norberto Gavioli, An ultimately periodic chain in the integral Lie ring of partitions, J. Algebr. Comb. (2024). See p. 11.

Thomas M. A. Fink, Emmanuel Barillot, and Sebastian E. Ahnert, Dynamics of network motifs, 2006.

Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 800

Formula

a(n) = A000070(n) - 1, n >= 1.

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 + 11*Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Oct 25 2016

G.f.: -1/(1 - x) + (1/(1 - x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Dec 25 2016

Maple

a:= n-> add(combinat[numbpart](k), k=1..n): seq(a(n), n=1..44); # Zerinvary Lajos, Jun 01 2008

Mathematica

Table[ Sum[ PartitionsP[k], {k, 1, n}], {n, 1, 45}]
(* or: *)
PartitionsP[Range[45]] // Accumulate (* Jean-François Alcover, Jun 19 2019 *)
CoefficientList[Series[(QPochhammer[x] - 1)/(x (x - 1) QPochhammer[x]), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 29 2022 *)

Prog

(PARI) a(n) = sum(k=1, n, numbpart(k)); \\ Michel Marcus, Jul 19 2023
(Python)
from sympy import partition
def A026905(n): return sum(partition(k) for k in range(1, n+1)) # Chai Wah Wu, Nov 23 2024

Crossrefs

Cf. A000041, A000070, A001477, A026906, A206437.

Rows sums of A133737, A137633, A137679.

Sequence in context: A014284 A118482 A281689 * A286272 A212147 A066778

Adjacent sequences: A026902 A026903 A026904 * A026906 A026907 A026908

Keyword

nonn

Author

Clark Kimberling

Extensions

Edited by N. J. A. Sloane, Jun 20 2015

Name clarified by Omar E. Pol, Apr 30 2022

Status

approved