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A138879 Sum of all parts of the last section of the set of partitions of n. 43
1, 3, 5, 11, 15, 31, 39, 71, 94, 150, 196, 308, 389, 577, 750, 1056, 1353, 1881, 2380, 3230, 4092, 5412, 6821, 8935, 11150, 14386, 17934, 22834, 28281, 35735, 43982, 55066, 67551, 83821, 102365, 126267, 153397, 188001, 227645, 277305, 334383 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums of the triangles A135010, A138121, A138151 and others related to the shell model of partitions (see A135010 and A138121).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A000041(n)*n - A000041(n-1)*(n-1) = A138880(n) + A000041(n-1).

a(n) = A066186(n) - A066186(n-1), for n>=1.

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi/(12*sqrt(2*n)) * (1 - (72 + 13*Pi^2) / (24*Pi*sqrt(6*n)) + (7/12 + 3/(2*Pi^2) + 217*Pi^2/6912)/n - (15*sqrt(3/2)/(16*Pi) + 115*Pi/(288*sqrt(6)) + 4069*Pi^3/(497664*sqrt(6)))/n^(3/2)). - Vaclav Kotesovec, Oct 21 2016, extended Jul 06 2019

G.f.: x*(1 - x)*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017

EXAMPLE

For n=6 the a(6)=31 because the parts of the last section of the set of partitions of 6 are (6),(3,3),(4,2),(2,2,2),(1),(1),(1),(1),(1),(1),(1), so the sum is a(6) = 6+3+3+4+2+2+2+2+1+1+1+1+1+1+1 = 31.

From Omar E. Pol, Aug 13 2013: (Start)

Illustration of initial terms:

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.   1    3      5        11         15           31

.

(End)

MAPLE

A066186 := proc(n) n*combinat[numbpart](n) ; end proc:

A138879 := proc(n) A066186(n)-A066186(n-1) ; end proc:

seq(A138879(n), n=1..80) ; # R. J. Mathar, Jan 27 2011

MATHEMATICA

Table[PartitionsP[n]*n - PartitionsP[n-1]*(n-1), {n, 1, 50}] (* Vaclav Kotesovec, Oct 21 2016 *)

PROG

(PARI) for(n=1, 50, print1(numbpart(n)*n - numbpart(n - 1)*(n - 1), ", ")) \\ Indranil Ghosh, Mar 19 2017

(Python)

from sympy.ntheory import npartitions

print[npartitions(n)*n - npartitions(n - 1)*(n - 1) for n in range(1, 51)] # Indranil Ghosh, Mar 19 2017

CROSSREFS

Cf. A000041, A066186, A133041, A135010, A138121, A138135 - A138138, A138151, A138880, A139100.

Sequence in context: A164053 A200176 A092929 * A318915 A322439 A018313

Adjacent sequences:  A138876 A138877 A138878 * A138880 A138881 A138882

KEYWORD

nonn

AUTHOR

Omar E. Pol, Apr 30 2008

EXTENSIONS

a(34) corrected by R. J. Mathar, Jan 27 2011

STATUS

approved

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Last modified February 27 12:47 EST 2020. Contains 332306 sequences. (Running on oeis4.)