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 A026905 Partial sums of the partition numbers A000041. 37

%I

%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

%N Partial sums of the partition numbers A000041.

%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

%H Thomas M. A. Fink, Emmanuel Barillot, and Sebastian E. Ahnert, <a href="http://www.tcm.phy.cam.ac.uk/~tmf20/PUBLICATIONS/dynamics_motifs.pdf">Dynamics of network motifs</a>, 2006.

%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(numbpart(k), k=1..n): seq(a(n), n=1..40); # _Zerinvary Lajos_, Jun 01 2008

%t Table[ Sum[ PartitionsP[k], {k, 1, n}], {n, 1, 45}]

%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

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