A117525 Total sum of parts of multiplicity 2 in all partitions of n.

0, 0, 1, 0, 3, 3, 7, 9, 20, 22, 44, 56, 90, 119, 186, 236, 355, 461, 651, 848, 1177, 1506, 2050, 2626, 3482, 4443, 5823, 7353, 9524, 11983, 15307, 19163, 24277, 30174, 37920, 46925, 58463, 72006, 89155, 109209, 134418, 163973, 200605, 243700, 296696, 358862

Offset: 0

Author

Vladeta Jovovic, Apr 26 2006

Keywords

• easy
• nonn

Comments

For m > 0, column m of A222730 is asymptotic to sqrt(3) * (2*m+1) * exp(Pi*sqrt(2*n/3)) / (2 * m^2 * (m+1)^2 * Pi^2) ~ 6 * (2*m+1) * n * p(n) / (m^2 * (m+1)^2 * Pi^2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, May 29 2018

Examples

			a(5) = 3 because the partitions of 5 that have parts with multiplicity 2 are [3,1,1] and [2,2,1] and the sum of those parts is 1+2 = 3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A103628.

Column k=2 of A222730. - Alois P. Heinz, Mar 03 2013

Programs

Maple

g:=(x^2/(1-x^2)^2-x^3/(1-x^3)^2)/Product(1-x^i,i=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..50); # Emeric Deutsch, May 13 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=2, [l[1], l[2]+l[1]*i], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2013

Mathematica

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[4]]; a[0] = a[1] = 0; Table[a[n], {n, 0, 50}]   (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

Formula

G.f. for total sum of parts of multiplicity m in all partitions of n is (x^m/(1-x^m)^2-x^(m+1)/(1-x^(m+1))^2)/Product(1-x^i,i=1..infinity).

a(n) ~ 5 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (72 * Pi^2). - Vaclav Kotesovec, May 29 2018

Extensions

More terms from Emeric Deutsch, May 13 2006