cp's OEIS Frontend

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

%I A066898 #58 Jan 08 2025 05:48:19
%S A066898 0,1,1,4,5,11,15,28,38,62,85,131,177,258,346,489,648,890,1168,1572,
%T A066898 2042,2699,3475,4532,5783,7446,9430,12017,15106,19073,23815,29827,
%U A066898 37011,46012,56765,70116,86033,105627,128962,157476,191359,232499,281286,340180,409871
%N A066898 Total number of even parts in all partitions of n.
%C A066898 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 A066898 From _Omar E. Pol_, Apr 06 2023: (Start)
%C A066898 Convolution of A000041 and A183063.
%C A066898 Convolution of A002865 and A362059.
%C A066898 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 A066898 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 A066898 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 A066898 a(n) = Sum_{k=1..floor(n/2)} tau(k)*numbpart(n-2*k). - _Vladeta Jovovic_, Jan 26 2002
%F A066898 a(n) = Sum_{k=0..floor(n/2)} k*A116482(n,k). - _Emeric Deutsch_, Feb 17 2006
%F A066898 G.f.: (Sum_{j>=1} x^(2*j)/(1-x^(2*j)))/(Product_{j>=1} (1-x^j)). - _Emeric Deutsch_, Feb 17 2006
%F A066898 a(n) = A066897(n) - A209423(n) = A006128(n) - A066897(n). - _Reinhard Zumkeller_, Mar 09 2012
%F A066898 a(n) = (A006128(n) - A209423(n))/2. - _Vaclav Kotesovec_, May 25 2018
%F A066898 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 A066898 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 A066898 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 A066898 := proc(n)
%p A066898     add(numtheory[tau](k)*combinat[numbpart](n-2*k),k=1..n/2) ;
%p A066898 end proc: # _R. J. Mathar_, Jun 18 2016
%t A066898 f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]
%t A066898 o[n_] := Sum[f[n, i], {i, 1, n, 2}]
%t A066898 e[n_] := Sum[f[n, i], {i, 2, n, 2}]
%t A066898 Table[o[n], {n, 1, 45}]  (* A066897 *)
%t A066898 Table[e[n], {n, 1, 45}]  (* A066898 *)
%t A066898 %% - %                   (* A209423 *)
%t A066898 (* _Clark Kimberling_, Mar 08 2012 *)
%t A066898 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 A066898 (Haskell)
%o A066898 a066898 = p 0 1 where
%o A066898    p e _             0 = e
%o A066898    p e k m | m < k     = 0
%o A066898            | otherwise = p (e + 1 - mod k 2) k (m - k) + p e (k + 1) m
%o A066898 -- _Reinhard Zumkeller_, Mar 09 2012
%o A066898 (Haskell)
%o A066898 a066898 = length . filter even . concat . ps 1 where
%o A066898    ps _ 0 = [[]]
%o A066898    ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
%o A066898 -- _Reinhard Zumkeller_, Jul 13 2013
%Y A066898 Cf. A000005, A000041, A002865, A006128, A066897, A116482, A183063, A206563, A209423, A362059.
%Y A066898 Column 2 of A206563.
%K A066898 easy,nonn
%O A066898 1,4
%A A066898 _Naohiro Nomoto_, Jan 24 2002
%E A066898 More terms from _Vladeta Jovovic_, Jan 26 2002