A066966 Total sum of even parts in all partitions of n.

0, 2, 2, 10, 12, 30, 40, 82, 110, 190, 260, 422, 570, 860, 1160, 1690, 2252, 3170, 4190, 5760, 7540, 10142, 13164, 17450, 22442, 29300, 37410, 48282, 61170, 78132, 98310, 124444, 155582, 195310, 242722, 302570, 373882, 462954, 569130, 700570, 856970

Offset

1,2

Comments

Partial sums of A206436. - Omar E. Pol, Mar 17 2012

From Omar E. Pol, Apr 02 2023: (Start)

Convolution of A000041 and A146076.

Convolution of A002865 and A271342.

a(n) is also the sum of all 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

Alois P. Heinz, Table of n, a(n) for n = 1..1000

George E. Andrews and Mircea Merca, A further look at the sum of the parts with the same parity in the partitions of n, Journal of Combinatorial Theory, Series A, Volume 203, 105849 (2024).

Formula

a(n) = 2*Sum_{k=1..floor{n/2)} sigma(k)*numbpart(n-2*k).

a(n) = Sum_{k=0..n} k*A113686(n,k). - Emeric Deutsch, Feb 20 2006

G.f.: Sum_{j>=1} (2jx^(2j)/(1-x^(2j)))/Product_{j>=1}(1-x^j). - Emeric Deutsch, Feb 20 2006

a(n) = A066186(n) - A066967(n). - Omar E. Pol, Mar 10 2012

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - Vaclav Kotesovec, May 29 2018

Example

a(4) = 10 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], the total sum of the even parts is 4+2+2+2 = 10.

Maple

g:=sum(2*j*x^(2*j)/(1-x^(2*j)), j=1..55)/product(1-x^j, j=1..55): gser:=series(g, x=0, 45): seq(coeff(gser, x^n), n=1..41);
# Emeric Deutsch, Feb 20 2006
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012

Mathematica

max = 50; g = Sum[2*j*x^(2*j)/(1 - x^(2*j)), {j, 1, max}]/Product[1 - x^j, {j, 1, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n], {n, 1, max - 1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)
Map[Total[Select[Flatten[IntegerPartitions[#]], EvenQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)

Prog

(PARI) a(n) = 2*sum(k=1, floor(n/2), sigma(k)*numbpart(n-2*k) ); \\ Joerg Arndt, Jan 24 2014

Crossrefs

Cf. A000041, A000203, A002865, A066186, A066897, A066898, A066967, A113686, A146076, A206436, A271342, A338156.

Sequence in context: A263053 A361835 A066965 * A132443 A048153 A015623

Adjacent sequences: A066963 A066964 A066965 * A066967 A066968 A066969

Keyword

nonn

Author

Vladeta Jovovic, Jan 26 2002

Extensions

More terms from Naohiro Nomoto and Sascha Kurz, Feb 07 2002

More terms from Emeric Deutsch, Feb 20 2006

Status

approved