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

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A066898 Total number of even parts in all partitions of n. 23
0, 1, 1, 4, 5, 11, 15, 28, 38, 62, 85, 131, 177, 258, 346, 489, 648, 890, 1168, 1572, 2042, 2699, 3475, 4532, 5783, 7446, 9430, 12017, 15106, 19073, 23815, 29827, 37011, 46012, 56765, 70116, 86033, 105627, 128962, 157476, 191359, 232499, 281286, 340180, 409871 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also sum of all even-indexed parts minus the sum of all odd-indexed parts, except the largest parts, of all partitions of n (cf. A206563). - Omar E. Pol, Feb 14 2012

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.

FORMULA

a(n) = Sum_{k=1..floor{n/2)} tau(k)*numbpart(n-2*k). - Vladeta Jovovic, Jan 26 2002

a(n) = sum(k*A116482(n,k), k=0..floor(n/2)). - Emeric Deutsch, Feb 17 2006

G.f.: sum(x^(2j)/(1-x^(2j)), j=1..infinity)/product((1-x^j), j=1..infinity). - Emeric Deutsch, Feb 17 2006

a(n) = A066897(n) - A209423(n) = A006128(n) - A066897(n). [Reinhard Zumkeller, Mar 09 2012]

a(n) = (A006128(n) - A209423(n))/2. - Vaclav Kotesovec, May 25 2018

a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(3*n/(2*Pi^2))) / (8*Pi*sqrt(2*n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018

EXAMPLE

a(5) = 5 because in all the partitions of 5, namely [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], we have a total of 0+1+1+0+2+1+0=5 even parts.

MAPLE

g:=sum(x^(2*j)/(1-x^(2*j)), j=1..60)/product((1-x^j), j=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=1..50); # Emeric Deutsch, Feb 17 2006

A066898 := proc(n)

    add(numtheory[tau](k)*combinat[numbpart](n-2*k), k=1..n/2) ;

end proc: # R. J. Mathar, Jun 18 2016

MATHEMATICA

f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]

o[n_] := Sum[f[n, i], {i, 1, n, 2}]

e[n_] := Sum[f[n, i], {i, 2, n, 2}]

Table[o[n], {n, 1, 45}]  (* A066897 *)

Table[e[n], {n, 1, 45}]  (* A066898 *)

%% - %                   (* A209423 *)

(* Clark Kimberling, Mar 08 2012 *)

a[n_] := Sum[DivisorSigma[0, k] PartitionsP[n - 2k], {k, 1, n/2}]; Table[a[n], {n, 1, 50}] (* Jean-Fran├žois Alcover, Aug 31 2016, after Vladeta Jovovic *)

PROG

(Haskell)

a066898 = p 0 1 where

   p e _             0 = e

   p e k m | m < k     = 0

           | otherwise = p (e + 1 - mod k 2) k (m - k) + p e (k + 1) m

-- Reinhard Zumkeller, Mar 09 2012

(Haskell)

a066898 = length . filter even . concat . ps 1 where

   ps _ 0 = [[]]

   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]

-- Reinhard Zumkeller, Jul 13 2013

CROSSREFS

Cf. A000041, A000005, A006128, A066897, A116482.

Column 2 of A206563. - Omar E. Pol, Feb 15 2012

Sequence in context: A053307 A076065 A176115 * A118143 A001350 A077238

Adjacent sequences:  A066895 A066896 A066897 * A066899 A066900 A066901

KEYWORD

easy,nonn

AUTHOR

Naohiro Nomoto, Jan 24 2002

EXTENSIONS

More terms from Vladeta Jovovic, Jan 26 2002

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 December 9 03:27 EST 2019. Contains 329872 sequences. (Running on oeis4.)