login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A229327 Total sum of 5th powers of parts in all partitions of n. 2
0, 1, 34, 279, 1370, 4775, 14196, 35745, 83486, 177120, 358710, 681316, 1257414, 2212343, 3811590, 6344760, 10381686, 16534989, 25994160, 39973360, 60802860, 90875412, 134507694, 196208405, 283895550, 405646460, 575437476, 807778980, 1126478494, 1556675935 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The bivariate g.f. for the partition statistic "sum of 5th powers the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^5}*x^k). The g.f. g at the Formula section has been obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

FORMULA

a(n) = Sum_{k=1..n} A066633(n,k) * k^5.

G.f.: g(x) = (Sum_{k>=1} k^5*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015

a(n) ~ 16*sqrt(3)/7 * exp(Pi*sqrt(2*n/3)) * n^2. - Vaclav Kotesovec, May 28 2018

MAPLE

b:= proc(n, i) option remember; `if`(n=0, [1, 0],

      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),

      ((g, h)-> g+h+[0, h[1]*i^5])(b(n, i-1), b(n-i, i)))))

    end:

a:= n-> b(n, n)[2]:

seq(a(n), n=0..40);

# second Maple program:

g := (sum(k^5*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

MATHEMATICA

Table[Total[Flatten[IntegerPartitions[n]]^5], {n, 0, 30}] (* Harvey P. Dale, Jun 24 2014 *)

(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k<n := T[n, k] = T[n-k, k] + PartitionsP[n-k]; T[_, _] = 0; a[n_] := Sum[T[n, k]*k^5, {k, 1, n}]; Array[a, 45, 0] (* Jean-Fran├žois Alcover, Dec 15 2016 *)

CROSSREFS

Column k=5 of A213191.

Sequence in context: A248076 A301543 A252999 * A209891 A027006 A156531

Adjacent sequences:  A229324 A229325 A229326 * A229328 A229329 A229330

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 20 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 19 20:41 EDT 2019. Contains 323410 sequences. (Running on oeis4.)