A325915 Total number of colors used in all colored integer partitions of n where all colors from an initial interval of the color palette are used and parts differ by size or by color.

0, 1, 3, 9, 25, 67, 176, 453, 1149, 2882, 7161, 17654, 43238, 105303, 255210, 615896, 1480771, 3548313, 8477415, 20199596, 48014369, 113879450, 269555798, 636875077, 1502195104, 3537705916, 8319377813, 19537936874, 45827441193, 107366261405, 251268532266

Offset: 0

Author

Alois P. Heinz, Sep 08 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2807

Crossrefs

Cf. A000009, A304969, A308680.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
g:= proc(n) option remember; `if`(n=0, [1, 0],
      (p-> p+[0, p[1]])(add(b(j)*g(n-j), j=1..n)))
    end:
a:= n-> g(n)[2]:
seq(a(n), n=0..32);

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j] Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
g[n_] := g[n] = If[n == 0, {1, 0}, Function[p, p + {0, p[[1]]}][Sum[b[j] g[n - j], {j, 1, n}]]];
a[n_] := g[n][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A308680(n,k).

a(n) ~ c * d^n * n, where d = 2.26562663992642295791262530033324290454663... is the root of the equation QPochhammer[-1, 1/d] = 4 and c = 0.1771510533646387556482103930322780317974659818141571819... - Vaclav Kotesovec, Sep 18 2019