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

Author

Clark Kimberling

Keywords

• nonn

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

• 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

Crossrefs

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

Rows sums of A133737, A137633, A137679.

Programs

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 *)

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

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

Extensions

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

Name clarified by Omar E. Pol, Apr 30 2022