A066898 Total number of even parts in all partitions of n.

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

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

From Omar E. Pol, Apr 06 2023: (Start)

Convolution of A000041 and A183063.

Convolution of A002865 and A362059.

a(n) is also the total number of even divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned even divisors are also all even parts of all partitions of n. (End)

Links

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

P. J. Grabner and 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-François 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. A000005, A000041, A002865, A006128, A066897, A116482, A183063, A206563, A209423, A362059.

Column 2 of A206563.

Sequence in context: A053307 A076065 A176115 * A118143 A001350 A378621

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