OFFSET
0,5
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
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
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Apr 26 2006
EXTENSIONS
More terms from Emeric Deutsch, May 13 2006
STATUS
approved