

A026905


Partial sums of the partition numbers A000041.


38



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
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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


LINKS

Table of n, a(n) for n=1..40.
Thomas M. A. Fink, Emmanuel Barillot, and Sebastian E. Ahnert, Dynamics of network motifs, 2006.
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(numbpart(k), k=1..n): seq(a(n), n=1..40); # Zerinvary Lajos, Jun 01 2008


MATHEMATICA

Table[ Sum[ PartitionsP[k], {k, 1, n}], {n, 1, 45}]
(* or: *)
PartitionsP[Range[45]] // Accumulate (* JeanFrançois Alcover, Jun 19 2019 *)


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


STATUS

approved



