OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Margaret Archibald, A. Blecher, C. Brennan, A. Knopfmacher and T. Mansour, Partitions according to multiplicities and part sizes, Australasian Journal of Combinatorics, Volume 66(1) (2016), Pages 104-119.
Ljuben Mutafchiev, On the Largest Part Size and Its Multiplicity of a Random Integer Partition, arXiv:1712.03233 [math.PR], 2017.
FORMULA
G.f.: Sum_{n>=1} (n*x^n/(1-x^n))*Product_{k=1..n} 1/(1-x^k).
EXAMPLE
Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4]; thus a(4) = 4*1 + 1*2 + 2*2 + 1*3 + 1*4 = 17.
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0],
`if`(i<1, [0$2], b(n, i-1, t) +add((l->`if`(t, l,
l+[0, l[1]*i*j]))(b(n-i*j, i-1, true)), j=1..n/i)))
end:
a:= n-> b(n$2, false)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Jan 29 2014
MATHEMATICA
f[n_] := Block[{c = 2n, k = 2, p = IntegerPartitions[n]}, m = Max @@@ p; l = Length[p]; While[k < l, c = c + m[[k]]*Count[p[[k]], m[[k]]]; k++ ]; If[n == 1, 1, c]]; Table[ f[n], {n, 41}] (* Robert G. Wilson v, Feb 18 2004, updated by Jean-François Alcover, Jan 29 2014 *)
nmax = 50; CoefficientList[Series[Sum[n*x^n/(1-x^n) * Product[1/(1 - x^k), {k, 1, n}], {n, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 06 2019 *)
Join[{0}, Table[Total[Flatten[First[Split[#]]&/@IntegerPartitions[n]]], {n, 50}]] (* Harvey P. Dale, Oct 29 2019 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Feb 16 2004
EXTENSIONS
More terms from Robert G. Wilson v, Feb 18 2004
STATUS
approved