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A026905
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Partial sums of the partition numbers A000041 of the positive integers.
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60
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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
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OFFSET
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1,2
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COMMENTS
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Equivalently, a(n) = number of sums S of positive integers satisfying S <= n.
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
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LINKS
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FORMULA
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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
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MAPLE
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a:= n-> add(combinat[numbpart](k), k=1..n): seq(a(n), n=1..44); # Zerinvary Lajos, Jun 01 2008
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MATHEMATICA
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Table[ Sum[ PartitionsP[k], {k, 1, n}], {n, 1, 45}]
(* or: *)
CoefficientList[Series[(QPochhammer[x] - 1)/(x (x - 1) QPochhammer[x]), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 29 2022 *)
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PROG
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(PARI) a(n) = sum(k=1, n, numbpart(k)); \\ Michel Marcus, Jul 19 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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