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%I #66 Apr 11 2024 10:33:39
%S 1,3,6,11,18,29,44,66,96,138,194,271,372,507,683,914,1211,1596,2086,
%T 2713,3505,4507,5762,7337,9295,11731,14741,18459,23024,28628,35470,
%U 43819,53962,66272,81155,99132,120769,146784,177969,215307,259890,313064,376325,451500
%N Partial sums of the partition numbers A000041 of the positive integers.
%C Equivalently, a(n) = number of sums S of positive integers satisfying S <= n.
%C Equivalently, first differences give A000041. - _Jacques ALARDET_, Aug 04 2008, Aug 15 2008
%C For the partial sums of the partitions numbers of nonnegative integers A001477 see A000070. - _Omar E. Pol_, Nov 12 2011
%C Also number of parts in all regions of n that contain 1 as a part (Cf. A206437). - _Omar E. Pol_, Mar 11 2012
%C Also the number of graph minors of the path graph P_n (not counting the null graph). - _Eric W. Weisstein_, Apr 29 2022
%H Riccardo Aragona, Roberto Civino, and Norberto Gavioli, <a href="https://doi.org/10.1007/s10801-024-01318-x">An ultimately periodic chain in the integral Lie ring of partitions</a>, J. Algebr. Comb. (2024). See p. 11.
%H Thomas M. A. Fink, Emmanuel Barillot, and Sebastian E. Ahnert, <a href="https://web.archive.org/web/20210427075631/http://www.tcm.phy.cam.ac.uk/~tmf20/PUBLICATIONS/dynamics_motifs.pdf">Dynamics of network motifs</a>, 2006.
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 23.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=800">Encyclopedia of Combinatorial Structures 800</a>
%F a(n) = A000070(n) - 1, n >= 1.
%F 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
%F G.f.: -1/(1 - x) + (1/(1 - x))*Product_{k>=1} 1/(1 - x^k). - _Ilya Gutkovskiy_, Dec 25 2016
%p a:= n-> add(combinat[numbpart](k), k=1..n): seq(a(n), n=1..44); # _Zerinvary Lajos_, Jun 01 2008
%t Table[ Sum[ PartitionsP[k], {k, 1, n}], {n, 1, 45}]
%t (* or: *)
%t PartitionsP[Range[45]] // Accumulate (* _Jean-François Alcover_, Jun 19 2019 *)
%t CoefficientList[Series[(QPochhammer[x] - 1)/(x (x - 1) QPochhammer[x]), {x, 0, 20}], x] (* _Eric W. Weisstein_, Apr 29 2022 *)
%o (PARI) a(n) = sum(k=1, n, numbpart(k)); \\ _Michel Marcus_, Jul 19 2023
%Y Cf. A000041, A000070, A001477, A026906, A206437.
%Y Rows sums of A133737, A137633, A137679.
%K nonn
%O 1,2
%A _Clark Kimberling_
%E Edited by _N. J. A. Sloane_, Jun 20 2015
%E Name clarified by _Omar E. Pol_, Apr 30 2022
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