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A066898
Total number of even parts in all partitions of n.
24
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
KEYWORD
easy,nonn
AUTHOR
Naohiro Nomoto, Jan 24 2002
EXTENSIONS
More terms from Vladeta Jovovic, Jan 26 2002
STATUS
approved