A066898 - OEIS

%I #58 Jan 08 2025 05:48:19

%S 0,1,1,4,5,11,15,28,38,62,85,131,177,258,346,489,648,890,1168,1572,

%T 2042,2699,3475,4532,5783,7446,9430,12017,15106,19073,23815,29827,

%U 37011,46012,56765,70116,86033,105627,128962,157476,191359,232499,281286,340180,409871

%N Total number of even parts in all partitions of n.

%C 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

%C From _Omar E. Pol_, Apr 06 2023: (Start)

%C Convolution of A000041 and A183063.

%C Convolution of A002865 and A362059.

%C 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)

%H Vaclav Kotesovec, <a href="/A066898/b066898.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)

%H P. J. Grabner and A. Knopfmacher, <a href="https://www.math.tugraz.at/fosp/pdfs/tugraz_0087.pdf">Analysis of some new partition statistics</a>, Ramanujan J., 12, 2006, 439-454.

%F a(n) = Sum_{k=1..floor(n/2)} tau(k)*numbpart(n-2*k). - _Vladeta Jovovic_, Jan 26 2002

%F a(n) = Sum_{k=0..floor(n/2)} k*A116482(n,k). - _Emeric Deutsch_, Feb 17 2006

%F G.f.: (Sum_{j>=1} x^(2*j)/(1-x^(2*j)))/(Product_{j>=1} (1-x^j)). - _Emeric Deutsch_, Feb 17 2006

%F a(n) = A066897(n) - A209423(n) = A006128(n) - A066897(n). - _Reinhard Zumkeller_, Mar 09 2012

%F a(n) = (A006128(n) - A209423(n))/2. - _Vaclav Kotesovec_, May 25 2018

%F 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

%e 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.

%p 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

%p A066898 := proc(n)

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

%p end proc: # _R. J. Mathar_, Jun 18 2016

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

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

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

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

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

%t %% - %                   (* A209423 *)

%t (* _Clark Kimberling_, Mar 08 2012 *)

%t 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_ *)

%o (Haskell)

%o a066898 = p 0 1 where

%o    p e _             0 = e

%o    p e k m | m < k     = 0

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

%o -- _Reinhard Zumkeller_, Mar 09 2012

%o (Haskell)

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

%o    ps _ 0 = [[]]

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

%o -- _Reinhard Zumkeller_, Jul 13 2013

%Y Cf. A000005, A000041, A002865, A006128, A066897, A116482, A183063, A206563, A209423, A362059.

%Y Column 2 of A206563.

%K easy,nonn

%O 1,4

%A _Naohiro Nomoto_, Jan 24 2002

%E More terms from _Vladeta Jovovic_, Jan 26 2002